CINXE.COM

<!DOCTYPE html> <html lang="en" xmlns:fb="http://www.facebook.com/2008/fbml" class="wf-loading"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb# academia: https://ogp.me/ns/fb/academia#"> <meta charset="utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <title>Masato Kobayashi - Academia.edu</title>  <link href="//a.academia-assets.com/images/favicons/favicon-production.ico" rel="shortcut icon" type="image/vnd.microsoft.icon"> <link rel="apple-touch-icon" sizes="57x57" href="//a.academia-assets.com/images/favicons/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon" sizes="60x60" href="//a.academia-assets.com/images/favicons/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon" sizes="72x72" href="//a.academia-assets.com/images/favicons/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon" sizes="76x76" href="//a.academia-assets.com/images/favicons/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon" sizes="114x114" href="//a.academia-assets.com/images/favicons/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon" sizes="120x120" href="//a.academia-assets.com/images/favicons/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon" sizes="144x144" href="//a.academia-assets.com/images/favicons/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon" sizes="152x152" href="//a.academia-assets.com/images/favicons/apple-touch-icon-152x152.png"> <link rel="apple-touch-icon" sizes="180x180" href="//a.academia-assets.com/images/favicons/apple-touch-icon-180x180.png"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-194x194.png" sizes="194x194"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/android-chrome-192x192.png" sizes="192x192"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-16x16.png" sizes="16x16"> <link rel="manifest" href="//a.academia-assets.com/images/favicons/manifest.json"> <meta name="msapplication-TileColor" content="#2b5797"> <meta name="msapplication-TileImage" content="//a.academia-assets.com/images/favicons/mstile-144x144.png"> <meta name="theme-color" content="#ffffff"> <script> window.performance && window.performance.measure && window.performance.measure("Time To First Byte", "requestStart", "responseStart"); </script> <script> (function() { if (!window.URLSearchParams || !window.history || !window.history.replaceState) { return; } var searchParams = new URLSearchParams(window.location.search); var paramsToDelete = [ 'fs', 'sm', 'swp', 'iid', 'nbs', 'rcc', // related content category 'rcpos', // related content carousel position 'rcpg', // related carousel page 'rchid', // related content hit id 'f_ri', // research interest id, for SEO tracking 'f_fri', // featured research interest, for SEO tracking (param key without value) 'f_rid', // from research interest directory for SEO tracking 'f_loswp', // from research interest pills on LOSWP sidebar for SEO tracking 'rhid', // referrring hit id ]; if (paramsToDelete.every((key) => searchParams.get(key) === null)) { return; } paramsToDelete.forEach((key) => { searchParams.delete(key); }); var cleanUrl = new URL(window.location.href); cleanUrl.search = searchParams.toString(); history.replaceState({}, document.title, cleanUrl); })(); </script> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "profiles/works", 'action': "summary", 'controller_action': 'profiles/works#summary', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script type="text/javascript"> window.sendUserTiming = function(timingName) { if (!(window.performance && window.performance.measure)) return; var entries = window.performance.getEntriesByName(timingName, "measure"); if (entries.length !== 1) return; var timingValue = Math.round(entries[0].duration); gtag('event', 'timing_complete', { name: timingName, value: timingValue, event_category: 'User-centric', }); }; window.sendUserTiming("Time To First Byte"); </script> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="5xIOxAH74C4I74IFEqtaBKbLwUaIv3O6rzOU3vZpFHfPW4FZ9XNX8UQ300il3zEoa03P5/1JvYBsOH4VTpbb5Q==" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/wow-77f7b87cb1583fc59aa8f94756ebfe913345937eb932042b4077563bebb5fb4b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/social/home-1c712297ae3ac71207193b1bae0ecf1aae125886850f62c9c0139dd867630797.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/heading-b2b823dd904da60a48fd1bfa1defd840610c2ff414d3f39ed3af46277ab8df3b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-3cea6e0ad4715ed965c49bfb15dedfc632787b32ff6d8c3a474182b231146ab7.css" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/common-10fa40af19d25203774df2d4a03b9b5771b45109c2304968038e88a81d1215c5.css" /> <meta name="author" content="masato kobayashi" /> <meta name="description" content="Masato Kobayashi: 1 Following, 13 Research papers. Research interests: Pure Mathematics, Discrete Mathematics, and Mathematical Sciences." /> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs" /> <script> var $controller_name = 'works'; var $action_name = "summary"; var $rails_env = 'production'; var $app_rev = '7f20a93c141b9b490992296828f53069361587f0'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.Aedu = { hit_data: null }; window.Aedu.SiteStats = {"premium_universities_count":15276,"monthly_visitors":"113 million","monthly_visitor_count":113784677,"monthly_visitor_count_in_millions":113,"user_count":277499391,"paper_count":55203019,"paper_count_in_millions":55,"page_count":432000000,"page_count_in_millions":432,"pdf_count":16500000,"pdf_count_in_millions":16}; window.Aedu.serverRenderTime = new Date(1732758600000); window.Aedu.timeDifference = new Date().getTime() - 1732758600000; window.Aedu.isUsingCssV1 = false; window.Aedu.enableLocalization = true; window.Aedu.activateFullstory = false; window.Aedu.serviceAvailability = { status: {"attention_db":"on","bibliography_db":"on","contacts_db":"on","email_db":"on","indexability_db":"on","mentions_db":"on","news_db":"on","notifications_db":"on","offsite_mentions_db":"on","redshift":"on","redshift_exports_db":"on","related_works_db":"on","ring_db":"on","user_tests_db":"on"}, serviceEnabled: function(service) { return this.status[service] === "on"; }, readEnabled: function(service) { return this.serviceEnabled(service) || this.status[service] === "read_only"; }, }; window.Aedu.viewApmTrace = function() { // Check if x-apm-trace-id meta tag is set, and open the trace in APM // in a new window if it is. var apmTraceId = document.head.querySelector('meta[name="x-apm-trace-id"]'); if (apmTraceId) { var traceId = apmTraceId.content; // Use trace ID to construct URL, an example URL looks like: // https://app.datadoghq.com/apm/traces?query=trace_id%31298410148923562634 var apmUrl = 'https://app.datadoghq.com/apm/traces?query=trace_id%3A' + traceId; window.open(apmUrl, '_blank'); } }; </script>  <link href="https://fonts.googleapis.com/css?family=Roboto:100,100i,300,300i,400,400i,500,500i,700,700i,900,900i" rel="stylesheet"> <link href="//maxcdn.bootstrapcdn.com/font-awesome/4.3.0/css/font-awesome.min.css" rel="stylesheet"> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/libraries-a9675dcb01ec4ef6aa807ba772c7a5a00c1820d3ff661c1038a20f80d06bb4e4.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/academia-296162c7af6fd81dcdd76f1a94f1fad04fb5f647401337d136fe8b68742170b1.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system_legacy-056a9113b9a0f5343d013b29ee1929d5a18be35fdcdceb616600b4db8bd20054.css" /> <script src="//a.academia-assets.com/assets/webpack_bundles/runtime-bundle-005434038af4252ca37c527588411a3d6a0eabb5f727fac83f8bbe7fd88d93bb.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/webpack_libraries_and_infrequently_changed.wjs-bundle-7734a8b54dedb8fefe75ef934203d0ef41d066c1c5cfa6ba2a13dd7ec8bce449.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/core_webpack.wjs-bundle-9eaa790b1139e2d88eb4ac931dcbc6d7cea4403a763db5cc5a695dd2eaa34151.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/sentry.wjs-bundle-5fe03fddca915c8ba0f7edbe64c194308e8ce5abaed7bffe1255ff37549c4808.js"></script> <script> jade = window.jade || {}; jade.helpers = window.$h; jade._ = window._; </script>  <script id="tag-manager-head-root">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer_old','GTM-5G9JF7Z');</script>  <script> window.gptadslots = []; window.googletag = window.googletag || {}; window.googletag.cmd = window.googletag.cmd || []; </script> <script type="text/javascript"> // TODO(jacob): This should be defined, may be rare load order problem. // Checking if null is just a quick fix, will default to en if unset. // Better fix is to run this immedietely after I18n is set. if (window.I18n != null) { I18n.defaultLocale = "en"; I18n.locale = "en"; I18n.fallbacks = true; } </script> <link rel="canonical" href="https://independent.academia.edu/MasatoKobayashi4" /> </head>   <body class='c-profiles/works a-summary logged_out'>  <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script><script>window.fbAsyncLoad = function() { // Protection against double calling of this function if (window.FB) { return; } (function(d, s, id){ var js, fjs = d.getElementsByTagName(s)[0]; if (d.getElementById(id)) {return;} js = d.createElement(s); js.id = id; js.src = "//connect.facebook.net/en_US/sdk.js"; fjs.parentNode.insertBefore(js, fjs); }(document, 'script', 'facebook-jssdk')); } if (!window.defer_facebook) { // Autoload if not deferred window.fbAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.fbAsyncLoad(); }, 5000); }</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script><script>window.googleAsyncLoad = function() { // Protection against double calling of this function (function(d) { var js; var id = 'google-jssdk'; var ref = d.getElementsByTagName('script')[0]; if (d.getElementById(id)) { return; } js = d.createElement('script'); js.id = id; js.async = true; js.onload = loadGoogle; js.src = "https://accounts.google.com/gsi/client" ref.parentNode.insertBefore(js, ref); }(document)); } if (!window.defer_google) { // Autoload if not deferred window.googleAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.googleAsyncLoad(); }, 5000); }</script> <div id="tag-manager-body-root">  <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5G9JF7Z" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>   <script> window.addEventListener('load', function() { if (document.querySelector('input[name="commit"]')) { document.querySelector('input[name="commit"]').addEventListener('click', function() { gtag('event', 'click', { event_category: 'button', event_label: 'Log In' }) }) } }); </script> </div> <script>var _comscore = _comscore || []; _comscore.push({ c1: "2", c2: "26766707" }); (function() { var s = document.createElement("script"), el = document.getElementsByTagName("script")[0]; s.async = true; s.src = (document.location.protocol == "https:" ? "https://sb" : "http://b") + ".scorecardresearch.com/beacon.js"; el.parentNode.insertBefore(s, el); })();</script><img src="https://sb.scorecardresearch.com/p?c1=2&c2=26766707&cv=2.0&cj=1" style="position: absolute; visibility: hidden" /> <div id='react-modal'></div> <div class='DesignSystem'> <a class='u-showOnFocus' href='#site'> Skip to main content </a> </div> <div id="upgrade_ie_banner" style="display: none;"><p>Academia.edu no longer supports Internet Explorer.</p><p>To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.getElementById('upgrade_ie_banner').style.display = 'block'; }</script> <div class="DesignSystem bootstrap ShrinkableNav"><div class="navbar navbar-default main-header"><div class="container-wrapper" id="main-header-container"><div class="container"><div class="navbar-header"><div class="nav-left-wrapper u-mt0x"><div class="nav-logo"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="visible-xs-inline-block" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hidden-xs" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a></div><div class="nav-search"><div class="SiteSearch-wrapper select2-no-default-pills"><form class="js-SiteSearch-form DesignSystem" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><i class="SiteSearch-icon fa fa-search u-fw700 u-positionAbsolute u-tcGrayDark"></i><input class="js-SiteSearch-form-input SiteSearch-form-input form-control" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" value="" /></form></div></div></div><div class="nav-right-wrapper pull-right"><ul class="NavLinks js-main-nav list-unstyled"><li class="NavLinks-link"><a class="js-header-login-url Button Button--inverseGray Button--sm u-mb4x" id="nav_log_in" rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="NavLinks-link u-p0x"><a class="Button Button--inverseGray Button--sm u-mb4x" rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li></ul><button class="hidden-lg hidden-md hidden-sm u-ml4x navbar-toggle collapsed" data-target=".js-mobile-header-links" data-toggle="collapse" type="button"><span class="icon-bar"></span><span class="icon-bar"></span><span class="icon-bar"></span></button></div></div><div class="collapse navbar-collapse js-mobile-header-links"><ul class="nav navbar-nav"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li><li class="u-borderColorGrayLight u-borderBottom1 js-mobile-nav-expand-trigger"><a href="#">more&nbsp<span class="caret"></span></a></li><li><ul class="js-mobile-nav-expand-section nav navbar-nav u-m0x collapse"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/about">About</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/press">Press</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://medium.com/@academia">Blog</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/documents">Papers</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/hiring"><i class="fa fa-briefcase"></i> We're Hiring!</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://support.academia.edu/"><i class="fa fa-question-circle"></i> Help Center</a></li><li class="js-mobile-nav-collapse-trigger u-borderColorGrayLight u-borderBottom1 dropup" style="display:none"><a href="#">less&nbsp<span class="caret"></span></a></li></ul></li></ul></div></div></div><script>(function(){ var $moreLink = $(".js-mobile-nav-expand-trigger"); var $lessLink = $(".js-mobile-nav-collapse-trigger"); var $section = $('.js-mobile-nav-expand-section'); $moreLink.click(function(ev){ ev.preventDefault(); $moreLink.hide(); $lessLink.show(); $section.collapse('show'); }); $lessLink.click(function(ev){ ev.preventDefault(); $moreLink.show(); $lessLink.hide(); $section.collapse('hide'); }); })() if ($a.is_logged_in() || false) { new Aedu.NavigationController({ el: '.js-main-nav', showHighlightedNotification: false }); } else { $(".js-header-login-url").attr("href", $a.loginUrlWithRedirect()); } Aedu.autocompleteSearch = new AutocompleteSearch({el: '.js-SiteSearch-form'});</script></div></div> <div id='site' class='fixed'> <div id="content" class="clearfix"> <script>document.addEventListener('DOMContentLoaded', function(){ var $dismissible = $(".dismissible_banner"); $dismissible.click(function(ev) { $dismissible.hide(); }); });</script> <script src="//a.academia-assets.com/assets/webpack_bundles/profile.wjs-bundle-533ab26e53b28997c1064a839643ef40d866d497c665279ed26b743aacb62042.js" defer="defer"></script><script>Aedu.rankings = { showPaperRankingsLink: false } $viewedUser = Aedu.User.set_viewed( {"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4","photo":"/images/s65_no_pic.png","has_photo":false,"is_analytics_public":false,"interests":[{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":37345,"name":"Discrete Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Mathematics"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory"}]} ); if ($a.is_logged_in() && $viewedUser.is_current_user()) { $('body').addClass('profile-viewed-by-owner'); } $socialProfiles = []</script><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://independent.academia.edu/MasatoKobayashi4","location":"/MasatoKobayashi4","scheme":"https","host":"independent.academia.edu","port":null,"pathname":"/MasatoKobayashi4","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="ProfileCheckPaperUpdate" data-props="{}" data-trace="false" data-dom-id="ProfileCheckPaperUpdate-react-component-0ae6c22d-ed1c-41fe-87ae-a1262c889783"></div> <div id="ProfileCheckPaperUpdate-react-component-0ae6c22d-ed1c-41fe-87ae-a1262c889783"></div> <div class="DesignSystem"><div class="onsite-ping" id="onsite-ping"></div></div><div class="profile-user-info DesignSystem"><div class="social-profile-container"><div class="left-panel-container"><div class="user-info-component-wrapper"><div class="user-summary-cta-container"><div class="user-summary-container"><div class="social-profile-avatar-container"><img class="profile-avatar u-positionAbsolute" border="0" alt="" src="//a.academia-assets.com/images/s200_no_pic.png" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Masato Kobayashi</h1><div class="affiliations-container fake-truncate js-profile-affiliations"></div></div></div><div class="sidebar-cta-container"><button class="ds2-5-button hidden profile-cta-button grow js-profile-follow-button" data-broccoli-component="user-info.follow-button" data-click-track="profile-user-info-follow-button" data-follow-user-fname="Masato" data-follow-user-id="216023629" data-follow-user-source="profile_button" data-has-google="false"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">add</span>Follow</button><button class="ds2-5-button hidden profile-cta-button grow js-profile-unfollow-button" data-broccoli-component="user-info.unfollow-button" data-click-track="profile-user-info-unfollow-button" data-unfollow-user-id="216023629"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">done</span>Following</button></div></div><div class="user-stats-container"><a><div class="stat-container js-profile-followers"><p class="label">Followers</p><p class="data">0</p></div></a><a><div class="stat-container js-profile-followees" data-broccoli-component="user-info.followees-count" data-click-track="profile-expand-user-info-following"><p class="label">Following</p><p class="data">1</p></div></a><span><div class="stat-container"><p class="label"><span class="js-profile-total-view-text">Public Views</span></p><p class="data"><span class="js-profile-view-count"></span></p></div></span></div><div class="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="216023629" href="https://www.academia.edu/Documents/in/Pure_Mathematics"><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://independent.academia.edu/MasatoKobayashi4","location":"/MasatoKobayashi4","scheme":"https","host":"independent.academia.edu","port":null,"pathname":"/MasatoKobayashi4","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Pure Mathematics"]}" data-trace="false" data-dom-id="Pill-react-component-531b00a8-9153-42f5-a2ed-e79dee803ce0"></div> <div id="Pill-react-component-531b00a8-9153-42f5-a2ed-e79dee803ce0"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="216023629" href="https://www.academia.edu/Documents/in/Discrete_Mathematics"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Discrete Mathematics"]}" data-trace="false" data-dom-id="Pill-react-component-b2854178-0cd0-4577-a535-395b771b7dc8"></div> <div id="Pill-react-component-b2854178-0cd0-4577-a535-395b771b7dc8"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="216023629" href="https://www.academia.edu/Documents/in/Mathematical_Sciences"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Mathematical Sciences"]}" data-trace="false" data-dom-id="Pill-react-component-88de8136-d16d-452c-b5fc-4f7fb631e911"></div> <div id="Pill-react-component-88de8136-d16d-452c-b5fc-4f7fb631e911"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="216023629" href="https://www.academia.edu/Documents/in/Number_Theory"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Number Theory"]}" data-trace="false" data-dom-id="Pill-react-component-737f4be2-e74d-4b7d-820f-fe3ede718dd8"></div> <div id="Pill-react-component-737f4be2-e74d-4b7d-820f-fe3ede718dd8"></div> </a></div></div></div></div><div class="right-panel-container"><div class="user-content-wrapper"><div class="uploads-container" id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Masato Kobayashi</h3></div><div class="js-work-strip profile--work_container" data-work-id="95559998"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559998/Schubert_Numbers"><img alt="Research paper thumbnail of Schubert Numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/97707786/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559998/Schubert_Numbers">Schubert Numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This thesis discusses intersections of the Schubert varieties in the flag variety associated to a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4caf13a14dae553799c5e3bcba915cdd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707786,"asset_id":95559998,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559998"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559998"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559998; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559998]").text(description); $(".js-view-count[data-work-id=95559998]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559998; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559998']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559998, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4caf13a14dae553799c5e3bcba915cdd" } } $('.js-work-strip[data-work-id=95559998]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559998,"title":"Schubert Numbers","translated_title":"","metadata":{"abstract":"This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders","publisher":"TRACE: Tennessee Research and Creative Exchange","publication_date":{"day":1,"month":5,"year":2010,"errors":{}}},"translated_abstract":"This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders","internal_url":"https://www.academia.edu/95559998/Schubert_Numbers","translated_internal_url":"","created_at":"2023-01-23T14:01:44.369-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707786,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707786/thumbnails/1.jpg","file_name":"268764458.pdf","download_url":"https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Schubert_Numbers.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707786/268764458-libre.pdf?1674512167=\u0026response-content-disposition=attachment%3B+filename%3DSchubert_Numbers.pdf\u0026Expires=1732762200\u0026Signature=LdU5SPOZb9S6vuw8lkb96BCciCz0zxkleedwf1a6AL5kcw53MDfKIgPKYJdjlmsaHK0tuSQnpp9bfTbMfmUyCUFCn8dRgUR7LJ6s3VR2x6OsLlL60BUzLgQ-2rYd0te068oKylbtYmU3AKqaFu9HrAcpFHq0CLNXqL2OuxhIbT0HH2u6YQaJSoIqb04wWWr0KyVW1aZBjaOK~fj3mwEous-babLLQ10MGLweS3k2dnWT8jx3mEhyzOaKxBWzFB49NEJAOWs~x3qsXMFcd96TkNqdVzFGJsvoTc6hjerGhS4WVWkzaZTk7~fSDh6y7-~TTwLmj4-oWc~zEQtnaWWjBQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Schubert_Numbers","translated_slug":"","page_count":71,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707786,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707786/thumbnails/1.jpg","file_name":"268764458.pdf","download_url":"https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Schubert_Numbers.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707786/268764458-libre.pdf?1674512167=\u0026response-content-disposition=attachment%3B+filename%3DSchubert_Numbers.pdf\u0026Expires=1732762200\u0026Signature=LdU5SPOZb9S6vuw8lkb96BCciCz0zxkleedwf1a6AL5kcw53MDfKIgPKYJdjlmsaHK0tuSQnpp9bfTbMfmUyCUFCn8dRgUR7LJ6s3VR2x6OsLlL60BUzLgQ-2rYd0te068oKylbtYmU3AKqaFu9HrAcpFHq0CLNXqL2OuxhIbT0HH2u6YQaJSoIqb04wWWr0KyVW1aZBjaOK~fj3mwEous-babLLQ10MGLweS3k2dnWT8jx3mEhyzOaKxBWzFB49NEJAOWs~x3qsXMFcd96TkNqdVzFGJsvoTc6hjerGhS4WVWkzaZTk7~fSDh6y7-~TTwLmj4-oWc~zEQtnaWWjBQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330395,"url":"https://core.ac.uk/download/268764458.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559996"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs"><img alt="Research paper thumbnail of Inequalities on Bruhat graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/97707819/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs">Inequalities on Bruhat graphs</a></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c2f89d98ce649fa6e9d0c89ccade0d67" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707819,"asset_id":95559996,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559996"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559996"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559996; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559996]").text(description); $(".js-view-count[data-work-id=95559996]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559996; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559996']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559996, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c2f89d98ce649fa6e9d0c89ccade0d67" } } $('.js-work-strip[data-work-id=95559996]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559996,"title":"Inequalities on Bruhat graphs","translated_title":"","metadata":{"grobid_abstract":"From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"grobid_abstract_attachment_id":97707819},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs","translated_internal_url":"","created_at":"2023-01-23T14:01:44.220-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707819,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707819/thumbnails/1.jpg","file_name":"1211.4305.pdf","download_url":"https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707819/1211.4305-libre.pdf?1674512140=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs.pdf\u0026Expires=1732762200\u0026Signature=cTr-4pT0rAqk462~XALwXTQoSArnl85ZqRl~vmI3vvtj8KO9UR4iqmKuu9InV4OMCr31KtG1InWxMg~h4w5kRRsAJf2Nzz4ZUO4auu7ldLrmzRPogtrOqMJSIokUpzaZvd-4BnCkuc~ggOSpNDw1EANzvI7W~jDCrl5nFYT36GdrFppgF7LD9nE7fD8Cs-2MOCo31OObc4A7HhmiXYKSeyqovDyedyIdYVk3Ub0xP9QgATf2hzR4XKMCa6ZBbG4iw97ylm3X4ctFlBC6eu~EC~i13F2ghjLWSi79DKzLG7OauNX9N8OqMOr~OANwlYKnKtpVLHg83XjFht1gC5a2-w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Inequalities_on_Bruhat_graphs","translated_slug":"","page_count":16,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707819,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707819/thumbnails/1.jpg","file_name":"1211.4305.pdf","download_url":"https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707819/1211.4305-libre.pdf?1674512140=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs.pdf\u0026Expires=1732762200\u0026Signature=cTr-4pT0rAqk462~XALwXTQoSArnl85ZqRl~vmI3vvtj8KO9UR4iqmKuu9InV4OMCr31KtG1InWxMg~h4w5kRRsAJf2Nzz4ZUO4auu7ldLrmzRPogtrOqMJSIokUpzaZvd-4BnCkuc~ggOSpNDw1EANzvI7W~jDCrl5nFYT36GdrFppgF7LD9nE7fD8Cs-2MOCo31OObc4A7HhmiXYKSeyqovDyedyIdYVk3Ub0xP9QgATf2hzR4XKMCa6ZBbG4iw97ylm3X4ctFlBC6eu~EC~i13F2ghjLWSi79DKzLG7OauNX9N8OqMOr~OANwlYKnKtpVLHg83XjFht1gC5a2-w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330394,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.3828\u0026rep=rep1\u0026type=pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559993"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups"><img alt="Research paper thumbnail of Bruhat order and graph structures of lower intervals in Coxeter groups" class="work-thumbnail" src="https://attachments.academia-assets.com/97707783/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups">Bruhat order and graph structures of lower intervals in Coxeter groups</a></div><div class="wp-workCard_item"><span>arXiv: Combinatorics</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-side...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="bc2b6a3236c97d82c35bbbd30522bf5b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707783,"asset_id":95559993,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559993"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559993"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559993; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559993]").text(description); $(".js-view-count[data-work-id=95559993]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559993; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559993']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559993, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "bc2b6a3236c97d82c35bbbd30522bf5b" } } $('.js-work-strip[data-work-id=95559993]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559993,"title":"Bruhat order and graph structures of lower intervals in Coxeter groups","translated_title":"","metadata":{"abstract":"We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.","publication_date":{"day":null,"month":null,"year":2018,"errors":{}},"publication_name":"arXiv: Combinatorics"},"translated_abstract":"We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.","internal_url":"https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups","translated_internal_url":"","created_at":"2023-01-23T14:01:44.085-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707783,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707783/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707783/1812.06266v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=WkMOBcSxj54WMQCptk1Wn0Z7A3UAORij8QRjW84m12aKizSGaYkasSzfZVLp6JXPRdNuE~AojVNvS9MGPrLTAMZSOkyCKIUOaLXQdBjGS6ebnN6Gw9h6xyFiEZ8DKOJF2TP5Wg-BpgRuuQrVQp3w7Aj3fEOM5GLefl4XIs4tOl1UKzHznh36rs-c2DQhBBj1tr-8BjkhaydNlCbpDhH-6H3x549En4HxhKdWGjoMDw4vIJIHBt9bAKKPpnHDm2i3xmrJqLWWXyH6WCF~UhSJs0hV-w5JH~lJibnykpX5S~jRm7O4pQdeYAdZmkSwkIgABnhPGBFwJr54yqLSGFaMCw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups","translated_slug":"","page_count":18,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707783,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707783/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707783/1812.06266v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=WkMOBcSxj54WMQCptk1Wn0Z7A3UAORij8QRjW84m12aKizSGaYkasSzfZVLp6JXPRdNuE~AojVNvS9MGPrLTAMZSOkyCKIUOaLXQdBjGS6ebnN6Gw9h6xyFiEZ8DKOJF2TP5Wg-BpgRuuQrVQp3w7Aj3fEOM5GLefl4XIs4tOl1UKzHznh36rs-c2DQhBBj1tr-8BjkhaydNlCbpDhH-6H3x549En4HxhKdWGjoMDw4vIJIHBt9bAKKPpnHDm2i3xmrJqLWWXyH6WCF~UhSJs0hV-w5JH~lJibnykpX5S~jRm7O4pQdeYAdZmkSwkIgABnhPGBFwJr54yqLSGFaMCw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707782,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707782/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707782/download_file","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707782/1812.06266v1-libre.pdf?1674512147=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=S-Wb0525EoJPDaqmNb73koFrqvQfjvBWoi3Upp6Lhp~TBBfJTwHF5UWMAwccItt2UWwB7kLrICsFj967WNCxtUe0AZsw5UPx5unw3mCYZCza1UlI1FQq6cZX5RyifxlQ58gWHx2hVQHz-FcpaFzLenkTKehM2AgtxFB6CETpQmqr07Te7NlZxxI9~bUtqYxNzc8DmDSLGxY8J5MDrQjQ4twSf8HFh1k30UyC1O703Wq5-XpKS2~RhhjYL2p-e5yfI-SRnjZ2BmsaAAxiv~CiPicfxXdA1RzFF5oDp0v~Ri~HYl4L6Fv7b7dnk9BFcM9MFwfe8r7jtGWf-lKmJsDrsA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":2696636,"name":"quotient","url":"https://www.academia.edu/Documents/in/quotient"}],"urls":[{"id":28330392,"url":"https://arxiv.org/pdf/1812.06266v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559991"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials"><img alt="Research paper thumbnail of On Determinants and Signed Bigrassmannian Polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/97707784/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials">On Determinants and Signed Bigrassmannian Polynomials</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9ba5c6680ed617b5e3c32589bababc6f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707784,"asset_id":95559991,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559991"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559991"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559991; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559991]").text(description); $(".js-view-count[data-work-id=95559991]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559991; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559991']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559991, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9ba5c6680ed617b5e3c32589bababc6f" } } $('.js-work-strip[data-work-id=95559991]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559991,"title":"On Determinants and Signed Bigrassmannian Polynomials","translated_title":"","metadata":{"abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).","publication_date":{"day":null,"month":null,"year":2014,"errors":{}}},"translated_abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).","internal_url":"https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_internal_url":"","created_at":"2023-01-23T14:01:43.954-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707784,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707784/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707784/_10_kobayashi-libre.pdf?1674512145=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=HeSV3mHlsAYCQkVd8H2dJezg8DVDD3hAJcGZ7PpdJ2WKRJ1CPL508CS90dEe8Ddmuw4dyZ3w0My0slb1~lat-YGpbEriaMGy2qC3N9Yy3oIJ6Le5ZHWHBpe~B4xHgkOn1PTAKx4cghOx3oe32F0BMc19x3F-ufsX6kRvFcADkEZ1iilg3v2h-UuwCzo8~xTg1btAGGIYMJgxAc3ueIuryHTSkJZQcFaUNcoqnVtEuBl4PM7YN9lCQOCbANQOrqNm5TzvdcJRO9MyzB0mRnFWjuKnukzaIR~RdAedpzLHxJMSvom0je-DgB0lBj3SX5puv0pwi-uPelGL15j5GLfoCg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_slug":"","page_count":14,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707784,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707784/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707784/_10_kobayashi-libre.pdf?1674512145=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=HeSV3mHlsAYCQkVd8H2dJezg8DVDD3hAJcGZ7PpdJ2WKRJ1CPL508CS90dEe8Ddmuw4dyZ3w0My0slb1~lat-YGpbEriaMGy2qC3N9Yy3oIJ6Le5ZHWHBpe~B4xHgkOn1PTAKx4cghOx3oe32F0BMc19x3F-ufsX6kRvFcADkEZ1iilg3v2h-UuwCzo8~xTg1btAGGIYMJgxAc3ueIuryHTSkJZQcFaUNcoqnVtEuBl4PM7YN9lCQOCbANQOrqNm5TzvdcJRO9MyzB0mRnFWjuKnukzaIR~RdAedpzLHxJMSvom0je-DgB0lBj3SX5puv0pwi-uPelGL15j5GLfoCg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707785,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707785/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707785/download_file","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707785/_10_kobayashi-libre.pdf?1674512152=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=BNOb3~oppVvMhJ4Xw-z~apSiS7eqHRViYjf6C7xz3U6QFASpT55sDiCr9SsdhtrTk7XoWBXNCyERh45eTeQ6bg6cAbl7epUCi5gPPaNLRWMO2iuqiB5wRzLD93oBrZvdWb3rTG1hXqV-ecqpnbpzOx~lvYec-pe7FAxL4giEDOgABtYt07B70SAQG16jOVTbHvtWWnkD3aKCvnEEKdhGUJFZe9cW1FPYiQb6KBh-N7AQwGXwOECTn3cNfWyUrjEEk-gg7811THLPDZVNX1qNHwDKz0uPjUs9L22RxGybXQv4VDk9i~LD5UrFvLmGj-fj9Kju7tWTrRRepmLH7EGBxw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330391,"url":"http://www.math.okayama-u.ac.jp/mjou/mjou57/_10_kobayashi.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559989"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups"><img alt="Research paper thumbnail of Two-sided structure of double cosets in Coxeter groups" class="work-thumbnail" src="https://attachments.academia-assets.com/97707780/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups">Two-sided structure of double cosets in Coxeter groups</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this article is to show one clear difference between single (left or right) cosets and...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c4e079e53e104e137f77f2923e71f955" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707780,"asset_id":95559989,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559989"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559989"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559989; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559989]").text(description); $(".js-view-count[data-work-id=95559989]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559989; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559989']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559989, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c4e079e53e104e137f77f2923e71f955" } } $('.js-work-strip[data-work-id=95559989]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559989,"title":"Two-sided structure of double cosets in Coxeter groups","translated_title":"","metadata":{"abstract":"The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.","publication_date":{"day":null,"month":null,"year":2011,"errors":{}}},"translated_abstract":"The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.","internal_url":"https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups","translated_internal_url":"","created_at":"2023-01-23T14:01:43.834-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707780,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707780/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707780/162-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=EJb2PkQhcyxxktkNYp2uszsVq0AgWR1nvaGe5pJbBsrlljLWcYNecIJoH1midQ86USCWmKVGtplG24sjW5s~t7GDgCLzbkwGBGn6IDeZ23CDPJvGliHl-rGy0oluPBVMwJbL1pxE~ehx2Nxzsg6q1sr9sLeg8zHRKwKs2xTRslDbi76sau9lhr4yrx7B25K6h2RQwXfoOW4mCU9uLVvEcF~-6kuvT7Hsd9DlsPIqV8kLLdsmZ85EcnqIwS2M~R343W0cqOLtBSc0ReUFmpgZzuslE6ZP0tPZduRd-cuSJzBznYpPtVdu9XAeTZK-hgXi0K5DakR07uuQJdLk-OnnRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Two_sided_structure_of_double_cosets_in_Coxeter_groups","translated_slug":"","page_count":9,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707780,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707780/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707780/162-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=EJb2PkQhcyxxktkNYp2uszsVq0AgWR1nvaGe5pJbBsrlljLWcYNecIJoH1midQ86USCWmKVGtplG24sjW5s~t7GDgCLzbkwGBGn6IDeZ23CDPJvGliHl-rGy0oluPBVMwJbL1pxE~ehx2Nxzsg6q1sr9sLeg8zHRKwKs2xTRslDbi76sau9lhr4yrx7B25K6h2RQwXfoOW4mCU9uLVvEcF~-6kuvT7Hsd9DlsPIqV8kLLdsmZ85EcnqIwS2M~R343W0cqOLtBSc0ReUFmpgZzuslE6ZP0tPZduRd-cuSJzBznYpPtVdu9XAeTZK-hgXi0K5DakR07uuQJdLk-OnnRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707781,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707781/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707781/download_file","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707781/162-libre.pdf?1674512149=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=OIL7F9rIvOrjTTh3ADTFmsDRJDXyEBvcKGiWwsL0Bm4Zr-8f~ia7NcaGs98Z-1mFlJ9hOeODXi4h6jmAJhQzD5ktN6tqj~N28cSX3LufniLPqylDHvDNznhakXQfV~qU643LB-uibDvxImanzZLyI4adXfsboa-XnD-OqBZEeewlbRNFg2heFkrceIGv14ecwRbPphchggype6H38Fm0o9NnVeYTOI6vGk6USiOkyTlVr8CMYWil4rkgHUr4pawKnP5M7XaqyDP40Ji8UTU5YkVi07iO81f~Xm4qUGiYl4EWnppJ7TPm1DRW-3oJaiimw4-7Epe2qjmTJLCCn78pxw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330390,"url":"http://www.math.titech.ac.jp/~tosho/Preprints/pdf/162.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559987"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set"><img alt="Research paper thumbnail of Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set" class="work-thumbnail" src="https://attachments.academia-assets.com/97707778/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set">Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set</a></div><div class="wp-workCard_item"><span>The Electronic Journal of Combinatorics</span><span>, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Bigrassmannian permutations are known as permutations which have precisely one left descent and o...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c6cafc084e5d32a4c7b83927867ebd68" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707778,"asset_id":95559987,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559987"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559987"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559987; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559987]").text(description); $(".js-view-count[data-work-id=95559987]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559987; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559987']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559987, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c6cafc084e5d32a4c7b83927867ebd68" } } $('.js-work-strip[data-work-id=95559987]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559987,"title":"Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set","translated_title":"","metadata":{"abstract":"Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","publisher":"The Electronic Journal of Combinatorics","publication_date":{"day":null,"month":null,"year":2010,"errors":{}},"publication_name":"The Electronic Journal of Combinatorics"},"translated_abstract":"Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","internal_url":"https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set","translated_internal_url":"","created_at":"2023-01-23T14:01:43.698-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707778,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707778/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707778/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=Gj13uQ8UsC-7q60wxTBbjSr8r4wlE5rTeFxPm8bnbxaUtaTobhI~FmqEAfTk-NWPj3TcLxWN66ZEBtFke9a-krNYtU-SUqvaDBl-5pjjnrRx-rj8SUhRjslSOEaja2oZE0vWJfT6FN7syM7tQYhkyX7wI~j9FV~pMOU5tmhmHpyespLgqMpQIrGU188feCp0K0ERn3yDDtusfY~AyN80GybWHLobdQDm~KmbkM0qihkHFuVW0SCt8il4AjccTRKgywQ3v3dtkA2ioIXT7~MWBNJ5yhLjYPFKSHkyEzai2DPf~N36fp~FEAOCXll-Npj~9hJrb5gI~ovvpQbViM4SJA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set","translated_slug":"","page_count":8,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707778,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707778/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707778/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=Gj13uQ8UsC-7q60wxTBbjSr8r4wlE5rTeFxPm8bnbxaUtaTobhI~FmqEAfTk-NWPj3TcLxWN66ZEBtFke9a-krNYtU-SUqvaDBl-5pjjnrRx-rj8SUhRjslSOEaja2oZE0vWJfT6FN7syM7tQYhkyX7wI~j9FV~pMOU5tmhmHpyespLgqMpQIrGU188feCp0K0ERn3yDDtusfY~AyN80GybWHLobdQDm~KmbkM0qihkHFuVW0SCt8il4AjccTRKgywQ3v3dtkA2ioIXT7~MWBNJ5yhLjYPFKSHkyEzai2DPf~N36fp~FEAOCXll-Npj~9hJrb5gI~ovvpQbViM4SJA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707779,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707779/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707779/download_file","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707779/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=eGq0L9vMfXOD6CcReTDiK3FSYAq5uXAmZuRxzObf3HpDH2Jvuh0KtdMwVlx5XULV~~eFZscwqYawODf9dMPUTdybprUr45OBx9CpzAFxaPbj19KoYT6uQ3XwFDLG62Su-JY01XVrIP6waNfjobNn26~BNcvzaH6XjyQ3xuhZe5QocaXuyO2gNIz23COBhnMryf3CDs-W~q-~tl7WP60ZncisL4b8BeQ5b2dyzVHD1hiZOkQKZpx0~auJnUgKs2fpTlKSm22brHLQZ4gklL7RnFg-BtjX9RR8KX-kBD0YMknyGHBY3Gulua7ePwTYrd4HthIc3TkGg24aD4kZcoI9Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":3381241,"name":"bijection","url":"https://www.academia.edu/Documents/in/bijection"}],"urls":[{"id":28330389,"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1n27/pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559983"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices"><img alt="Research paper thumbnail of Weighted Counting of Inversions on Alternating Sign Matrices" class="work-thumbnail" src="https://attachments.academia-assets.com/97707840/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices">Weighted Counting of Inversions on Alternating Sign Matrices</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2019</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fb494141832b24b0936ad068f166acd5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707840,"asset_id":95559983,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559983"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559983"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559983; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559983]").text(description); $(".js-view-count[data-work-id=95559983]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559983; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559983']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559983, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fb494141832b24b0936ad068f166acd5" } } $('.js-work-strip[data-work-id=95559983]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559983,"title":"Weighted Counting of Inversions on Alternating Sign Matrices","translated_title":"","metadata":{"publisher":"Springer Science and Business Media LLC","grobid_abstract":"We generalize the author's formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).","publication_date":{"day":null,"month":null,"year":2019,"errors":{}},"publication_name":"Order","grobid_abstract_attachment_id":97707840},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices","translated_internal_url":"","created_at":"2023-01-23T14:01:43.385-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707840,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707840/thumbnails/1.jpg","file_name":"1904.pdf","download_url":"https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Weighted_Counting_of_Inversions_on_Alter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707840/1904-libre.pdf?1674512137=\u0026response-content-disposition=attachment%3B+filename%3DWeighted_Counting_of_Inversions_on_Alter.pdf\u0026Expires=1732762200\u0026Signature=YGf95hlhquu0XSk5NVSqTftx6-NRtb5BiY2K3f-beopX713mG8nLW6~AekMwEQOgYobGaTZBVYairqoTvuNHo7jnd-HOVSw1glPPS9E~RhSdOKIbXY-kg42cDwb4GbUV1h219vhrJAVqS76dqk1ykaY5bAh-4FGiVPKb7BccZCgTWfHhB0iIuEIiuiju3aY2CYtsc2RguooIViJa2eaGp6AU2mnl652fWNWVq6qGd0pL1uVxP1LCTtC-zZw7kU~itUPmGngPiyQbmcKXpOABp3YusnjaxgaW9W6vOjRVZZoZ20pG9unJYUtATp-CXI7ccnlGjjPaKGDdPdvFsRdnCA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices","translated_slug":"","page_count":19,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707840,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707840/thumbnails/1.jpg","file_name":"1904.pdf","download_url":"https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Weighted_Counting_of_Inversions_on_Alter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707840/1904-libre.pdf?1674512137=\u0026response-content-disposition=attachment%3B+filename%3DWeighted_Counting_of_Inversions_on_Alter.pdf\u0026Expires=1732762200\u0026Signature=YGf95hlhquu0XSk5NVSqTftx6-NRtb5BiY2K3f-beopX713mG8nLW6~AekMwEQOgYobGaTZBVYairqoTvuNHo7jnd-HOVSw1glPPS9E~RhSdOKIbXY-kg42cDwb4GbUV1h219vhrJAVqS76dqk1ykaY5bAh-4FGiVPKb7BccZCgTWfHhB0iIuEIiuiju3aY2CYtsc2RguooIViJa2eaGp6AU2mnl652fWNWVq6qGd0pL1uVxP1LCTtC-zZw7kU~itUPmGngPiyQbmcKXpOABp3YusnjaxgaW9W6vOjRVZZoZ20pG9unJYUtATp-CXI7ccnlGjjPaKGDdPdvFsRdnCA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":183513,"name":"Order","url":"https://www.academia.edu/Documents/in/Order"}],"urls":[{"id":28330387,"url":"http://link.springer.com/content/pdf/10.1007/s11083-019-09515-1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices"><img alt="Research paper thumbnail of A directed graph structure of alternating sign matrices" class="work-thumbnail" src="https://attachments.academia-assets.com/97707825/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices">A directed graph structure of alternating sign matrices</a></div><div class="wp-workCard_item"><span>Linear Algebra and its Applications</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="72567e8df1cd7bfefc341deae85b8654" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707825,"asset_id":95559981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559981; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559981]").text(description); $(".js-view-count[data-work-id=95559981]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559981; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559981']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559981, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "72567e8df1cd7bfefc341deae85b8654" } } $('.js-work-strip[data-work-id=95559981]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559981,"title":"A directed graph structure of alternating sign matrices","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, q-analogs (which we name qTNN and qSFL properties). As a by-product, we obtain a new kind of permutation statistic, the signed bigrassmannian statistics, using Dodgson's condensation on determinants.","publication_date":{"day":null,"month":null,"year":2017,"errors":{}},"publication_name":"Linear Algebra and its Applications","grobid_abstract_attachment_id":97707825},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices","translated_internal_url":"","created_at":"2023-01-23T14:01:43.260-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707825,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707825/thumbnails/1.jpg","file_name":"1903.pdf","download_url":"https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_directed_graph_structure_of_alternatin.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707825/1903-libre.pdf?1674512155=\u0026response-content-disposition=attachment%3B+filename%3DA_directed_graph_structure_of_alternatin.pdf\u0026Expires=1732762200\u0026Signature=YfFamCvJzSe5a-sBQf0aHK5gKsLOldDaITypzFgbT4RKTmd6T0R1TZ5nBVcNu3Xd5h4IYMUcyeNUMp~EUHHYXEcMeJJIPkHGtEx59v5gNKYO80jLq9jNQyGgtqe1vJOnAHUSDkj1MvXfzc7dQWTqbG8-sAqPBtGr0sfpFHVYfjUFf-P9Ge2ypCEvcNjgnhtNo4Ay7DIZzUAYDA14~g48AqKj3ULOlTUZe15rZg3PxdE10Ng-hUT1hYL7gD3HzgAwEWLUcHdrzpn2nA7kqfCpo3iSXkR6pzHTGvXv0dg1hZXtpXeaSL02I9VxJw2729Ooz0N9uTy~HQvNrJWVz3exSg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_directed_graph_structure_of_alternating_sign_matrices","translated_slug":"","page_count":27,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707825,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707825/thumbnails/1.jpg","file_name":"1903.pdf","download_url":"https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_directed_graph_structure_of_alternatin.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707825/1903-libre.pdf?1674512155=\u0026response-content-disposition=attachment%3B+filename%3DA_directed_graph_structure_of_alternatin.pdf\u0026Expires=1732762200\u0026Signature=YfFamCvJzSe5a-sBQf0aHK5gKsLOldDaITypzFgbT4RKTmd6T0R1TZ5nBVcNu3Xd5h4IYMUcyeNUMp~EUHHYXEcMeJJIPkHGtEx59v5gNKYO80jLq9jNQyGgtqe1vJOnAHUSDkj1MvXfzc7dQWTqbG8-sAqPBtGr0sfpFHVYfjUFf-P9Ge2ypCEvcNjgnhtNo4Ay7DIZzUAYDA14~g48AqKj3ULOlTUZe15rZg3PxdE10Ng-hUT1hYL7gD3HzgAwEWLUcHdrzpn2nA7kqfCpo3iSXkR6pzHTGvXv0dg1hZXtpXeaSL02I9VxJw2729Ooz0N9uTy~HQvNrJWVz3exSg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":48,"name":"Engineering","url":"https://www.academia.edu/Documents/in/Engineering"},{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559977"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set"><img alt="Research paper thumbnail of More combinatorics of Fulton's essential set" class="work-thumbnail" src="https://attachments.academia-assets.com/97707824/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set">More combinatorics of Fulton's essential set</a></div><div class="wp-workCard_item"><span>International Mathematical Forum</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c9a48d45c133a9e6c19335f92ae70c4c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707824,"asset_id":95559977,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559977"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559977"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559977; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559977]").text(description); $(".js-view-count[data-work-id=95559977]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559977; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559977']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559977, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c9a48d45c133a9e6c19335f92ae70c4c" } } $('.js-work-strip[data-work-id=95559977]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559977,"title":"More combinatorics of Fulton's essential set","translated_title":"","metadata":{"publisher":"Hikari, Ltd.","ai_title_tag":"Combinatorics of Fulton's Essential Set and Baxter Permutations","grobid_abstract":"We develop combinatorics of Fulton's essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret Eriksson-Linusson's characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of Bruhat-Ehresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"International Mathematical Forum","grobid_abstract_attachment_id":97707824},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set","translated_internal_url":"","created_at":"2023-01-23T14:01:42.895-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707824,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707824/thumbnails/1.jpg","file_name":"25baf70f8f7631bf25e03f00e2ab7ed7b5ec.pdf","download_url":"https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"More_combinatorics_of_Fultons_essential.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707824/25baf70f8f7631bf25e03f00e2ab7ed7b5ec-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DMore_combinatorics_of_Fultons_essential.pdf\u0026Expires=1732762200\u0026Signature=SDplmj9Y8UH5KXMgl3tQeXuLyu-YCoZ4HCJZaJbLea~JmRoahXlPzysKhqXTi1Pgd7jQVnCWUR-WwCrqNxCPOM4s205-ZCIhkfsHtQ73qRhu9nyHgrzKWxTv0oCUvq4l-MSvAZH5wCM-EQwEHs2qBCM47DkaOupk7w70S-bVhGKRS5RuBvzua6AM9pc-spv4GuUuS3jeOgaME8N6yqfNqwOSLBRxU4HHKUMVUbqbFQaGh6b4RlgjfXu4uZV3JwluqN-iEjm4eswThV7uCfg7SUErON7mksXN0-O8EHeiwksFHK2ktw3JIPt6ts8LyGaMnnSMVGpNSSKPLBaXgk2sYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"More_combinatorics_of_Fultons_essential_set","translated_slug":"","page_count":26,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707824,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707824/thumbnails/1.jpg","file_name":"25baf70f8f7631bf25e03f00e2ab7ed7b5ec.pdf","download_url":"https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"More_combinatorics_of_Fultons_essential.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707824/25baf70f8f7631bf25e03f00e2ab7ed7b5ec-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DMore_combinatorics_of_Fultons_essential.pdf\u0026Expires=1732762200\u0026Signature=SDplmj9Y8UH5KXMgl3tQeXuLyu-YCoZ4HCJZaJbLea~JmRoahXlPzysKhqXTi1Pgd7jQVnCWUR-WwCrqNxCPOM4s205-ZCIhkfsHtQ73qRhu9nyHgrzKWxTv0oCUvq4l-MSvAZH5wCM-EQwEHs2qBCM47DkaOupk7w70S-bVhGKRS5RuBvzua6AM9pc-spv4GuUuS3jeOgaME8N6yqfNqwOSLBRxU4HHKUMVUbqbFQaGh6b4RlgjfXu4uZV3JwluqN-iEjm4eswThV7uCfg7SUErON7mksXN0-O8EHeiwksFHK2ktw3JIPt6ts8LyGaMnnSMVGpNSSKPLBaXgk2sYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":14193,"name":"Philosophy of Property","url":"https://www.academia.edu/Documents/in/Philosophy_of_Property"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559973"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order"><img alt="Research paper thumbnail of Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order" class="work-thumbnail" src="https://attachments.academia-assets.com/97707814/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order">Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2010</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f0dbf1f44016e58ec97666d95af6bd29" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707814,"asset_id":95559973,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559973"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559973"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559973; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559973]").text(description); $(".js-view-count[data-work-id=95559973]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559973; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559973']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559973, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f0dbf1f44016e58ec97666d95af6bd29" } } $('.js-work-strip[data-work-id=95559973]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559973,"title":"Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order","translated_title":"","metadata":{"publisher":"Springer Nature","grobid_abstract":"In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","publication_date":{"day":null,"month":null,"year":2010,"errors":{}},"publication_name":"Order","grobid_abstract_attachment_id":97707814},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order","translated_internal_url":"","created_at":"2023-01-23T14:01:42.638-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707814,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707814/thumbnails/1.jpg","file_name":"1005.pdf","download_url":"https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumeration_of_Bigrassmannian_Permutatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707814/1005-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DEnumeration_of_Bigrassmannian_Permutatio.pdf\u0026Expires=1732762200\u0026Signature=QSag-kw~VpYP0yfsg-9QfTpwkq9JNDgUBCeejAizmwoEhGfjpe-RlDJDzUWnnJ0Ito66b4Q6uIM81~Oz3rBjJXYOYA8~EkSYif-JBQyIPLVz0ctrmzu-30-COVijNUxJdRdswZk46p7RZwt-mb0QS47X73ZK3QCyBvZfAmusH5RtsxqJJhz~wJK~s0vtZjN8DRJ2ZEs1zHyO2znEajGDTvYtHllBnjwW7ScBTq5CCiOnfvSd8SpOvXGcvMhhxTcWLhuuGcTxYXBdiY08PVy56zYedS0dKfTFisHs5nppkLrWh34vDiUP34Ptt81~1-V7nKmldvTwe~XQ6RQblq4yFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order","translated_slug":"","page_count":7,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707814,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707814/thumbnails/1.jpg","file_name":"1005.pdf","download_url":"https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumeration_of_Bigrassmannian_Permutatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707814/1005-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DEnumeration_of_Bigrassmannian_Permutatio.pdf\u0026Expires=1732762200\u0026Signature=QSag-kw~VpYP0yfsg-9QfTpwkq9JNDgUBCeejAizmwoEhGfjpe-RlDJDzUWnnJ0Ito66b4Q6uIM81~Oz3rBjJXYOYA8~EkSYif-JBQyIPLVz0ctrmzu-30-COVijNUxJdRdswZk46p7RZwt-mb0QS47X73ZK3QCyBvZfAmusH5RtsxqJJhz~wJK~s0vtZjN8DRJ2ZEs1zHyO2znEajGDTvYtHllBnjwW7ScBTq5CCiOnfvSd8SpOvXGcvMhhxTcWLhuuGcTxYXBdiY08PVy56zYedS0dKfTFisHs5nppkLrWh34vDiUP34Ptt81~1-V7nKmldvTwe~XQ6RQblq4yFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":42156,"name":"Enumeration","url":"https://www.academia.edu/Documents/in/Enumeration"},{"id":150408,"name":"Symmetric group","url":"https://www.academia.edu/Documents/in/Symmetric_group"},{"id":183513,"name":"Order","url":"https://www.academia.edu/Documents/in/Order"},{"id":482747,"name":"Coxeter groups","url":"https://www.academia.edu/Documents/in/Coxeter_groups"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559942"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas"><img alt="Research paper thumbnail of Matrix representation of meet-irreducible discrete copulas" class="work-thumbnail" src="https://attachments.academia-assets.com/97707803/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas">Matrix representation of meet-irreducible discrete copulas</a></div><div class="wp-workCard_item"><span>Fuzzy Sets and Systems</span><span>, 2014</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="de59ac6c7d715973b371b7693cdf3d92" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707803,"asset_id":95559942,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559942"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559942"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559942; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559942]").text(description); $(".js-view-count[data-work-id=95559942]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559942; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559942']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559942, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "de59ac6c7d715973b371b7693cdf3d92" } } $('.js-work-strip[data-work-id=95559942]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559942,"title":"Matrix representation of meet-irreducible discrete copulas","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"Following Aguiló-Suñer-Torrens (2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we give characterizations of meet-irreducible discrete copulas from an order-theoretical aspect: we show that the set of all irreducible discrete copulas is a lattice in analogy with Nelsen andÚbeda-Flores (2005). Moreover, we clarify its lattice structure related to Kendall's τ and Spearman's ρ borrowing ideas from Coxeter groups.","publication_date":{"day":null,"month":null,"year":2014,"errors":{}},"publication_name":"Fuzzy Sets and Systems","grobid_abstract_attachment_id":97707803},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas","translated_internal_url":"","created_at":"2023-01-23T14:01:38.767-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707803,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707803/thumbnails/1.jpg","file_name":"2108.13564v1.pdf","download_url":"https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Matrix_representation_of_meet_irreducibl.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707803/2108.13564v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DMatrix_representation_of_meet_irreducibl.pdf\u0026Expires=1732762200\u0026Signature=eaUKAH45MRB1OEh2RgY5TwFR141DeSw3CEFuMMt2FstjsuFmQWz50~pUdYxtmPveHNMr-TrnBz4krhQzLN74kKjBuW90Hv5SpEe-YcFbMMzRzPd9s7Un~qSu8wMqS0rT1WoE-y2~Sp~y9VSLqJnFi2T2qHZAGADFMf9WlcgfECiU0M-o38QIeHbqXlS~tYYwEgBO1RI-3pWjSxXFizfMaT28orN2UWF4plnpVvMl5EhtiF3-Q1ebeR5sk4rql9N8rZY1Z3xyOgn~8G03YtGj1mogI~~Zg1cKIzgSBVVRyud3aexOHGwUR7Yty1HOFDaACKF0Ufsp4VAW8wlxGOHd5Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Matrix_representation_of_meet_irreducible_discrete_copulas","translated_slug":"","page_count":20,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707803,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707803/thumbnails/1.jpg","file_name":"2108.13564v1.pdf","download_url":"https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Matrix_representation_of_meet_irreducibl.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707803/2108.13564v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DMatrix_representation_of_meet_irreducibl.pdf\u0026Expires=1732762200\u0026Signature=eaUKAH45MRB1OEh2RgY5TwFR141DeSw3CEFuMMt2FstjsuFmQWz50~pUdYxtmPveHNMr-TrnBz4krhQzLN74kKjBuW90Hv5SpEe-YcFbMMzRzPd9s7Un~qSu8wMqS0rT1WoE-y2~Sp~y9VSLqJnFi2T2qHZAGADFMf9WlcgfECiU0M-o38QIeHbqXlS~tYYwEgBO1RI-3pWjSxXFizfMaT28orN2UWF4plnpVvMl5EhtiF3-Q1ebeR5sk4rql9N8rZY1Z3xyOgn~8G03YtGj1mogI~~Zg1cKIzgSBVVRyud3aexOHGwUR7Yty1HOFDaACKF0Ufsp4VAW8wlxGOHd5Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":115333,"name":"Analogy","url":"https://www.academia.edu/Documents/in/Analogy"},{"id":652157,"name":"Fuzzy Sets and Systems","url":"https://www.academia.edu/Documents/in/Fuzzy_Sets_and_Systems"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559894"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials"><img alt="Research paper thumbnail of Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/97707721/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials">Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="65409176c16466f98994a8aded64f879" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707721,"asset_id":95559894,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559894"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559894"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559894; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559894]").text(description); $(".js-view-count[data-work-id=95559894]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559894; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559894']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559894, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "65409176c16466f98994a8aded64f879" } } $('.js-work-strip[data-work-id=95559894]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559894,"title":"Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials","translated_title":"","metadata":{"abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)","publisher":"Department of Mathematics, Faculty of Science, Okayama University","publication_date":{"day":null,"month":null,"year":2015,"errors":{}}},"translated_abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)","internal_url":"https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_internal_url":"","created_at":"2023-01-23T14:00:13.123-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707721,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707721/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707721/32591052-libre.pdf?1674512153=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=J0c~FzsZDwSpmKOTmjw0chFevWTxzYTztq3iXMThuwRqe1RyjwByIDdKZKh0E9YptE4xtgmZgLgzizXgAERr9fIHdH9Y2wCfbRIRVOXlvO6VPsRUj52-gbGtFTYLtoYUbojNIPbzKXlyYdpD1H7TaioDIxQgl8MUqkCqOpkth0TWxbZHknLm9-LH~vtIez223GEmk6HUTZqCnFuYylndrKIvDKB4YDa~84vavZ38-odUdyQYrexFUj5ikNVXRydWJo7WGGixD1IPBz~3cPWJOw10TIGbULVxcsEfQ-arqpcxy0eJDKzWfbO63F8b~WjFf0Kahfo9U9flpPd6sDR3Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_slug":"","page_count":14,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707721,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707721/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707721/32591052-libre.pdf?1674512153=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=J0c~FzsZDwSpmKOTmjw0chFevWTxzYTztq3iXMThuwRqe1RyjwByIDdKZKh0E9YptE4xtgmZgLgzizXgAERr9fIHdH9Y2wCfbRIRVOXlvO6VPsRUj52-gbGtFTYLtoYUbojNIPbzKXlyYdpD1H7TaioDIxQgl8MUqkCqOpkth0TWxbZHknLm9-LH~vtIez223GEmk6HUTZqCnFuYylndrKIvDKB4YDa~84vavZ38-odUdyQYrexFUj5ikNVXRydWJo7WGGixD1IPBz~3cPWJOw10TIGbULVxcsEfQ-arqpcxy0eJDKzWfbO63F8b~WjFf0Kahfo9U9flpPd6sDR3Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707720,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707720/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707720/download_file","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707720/32591052-libre.pdf?1674512151=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=Fc~p8HnpBlie1beovhMVqhftdkNCHWZ5HW2VZtsS5Nks5Dn63EdYM-rQM-V92kzZY3c3kP41RynJtp~npHHLPlLP1mkLveta1DcOpl5HpbM76CrNB20sUDuEiB48bXXp7W4~5sK3mVjtExIvLNmwZXi4DJhSyWyZmCbRZoWSTkbdGioevpHGUR1HpjUcq2P2e-uzGjRCCwgPesJCi~QNPeW-r9onBtFT1bm5CemYLbqJtV2HGreZ79-mmT0PS6~GOFdfEI0p8H9O2j~fjntjOYyoXOaSSD2aTtdk4OyjBTtv0pEUTC3taZXFXUSW4uOJTJjaquW32P0QawmDy3Xsew__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":20428,"name":"Tournaments","url":"https://www.academia.edu/Documents/in/Tournaments"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":2508371,"name":"Macdonald polynomials","url":"https://www.academia.edu/Documents/in/Macdonald_polynomials"}],"urls":[{"id":28330345,"url":"https://core.ac.uk/download/32591052.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="87291472"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials"><img alt="Research paper thumbnail of Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/91541857/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials">Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials</a></div><div class="wp-workCard_item"><span>Journal of Combinatorial Theory, Series A</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4829426e06b1493a6c8129d03bad7447" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":91541857,"asset_id":87291472,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="87291472"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="87291472"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 87291472; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=87291472]").text(description); $(".js-view-count[data-work-id=87291472]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 87291472; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='87291472']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 87291472, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4829426e06b1493a6c8129d03bad7447" } } $('.js-work-strip[data-work-id=87291472]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":87291472,"title":"Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"Journal of Combinatorial Theory, Series A","grobid_abstract_attachment_id":91541857},"translated_abstract":null,"internal_url":"https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials","translated_internal_url":"","created_at":"2022-09-25T15:27:08.696-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":91541857,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/91541857/thumbnails/1.jpg","file_name":"1211.4305v1.pdf","download_url":"https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs_R_and_Kazh.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/91541857/1211.4305v1-libre.pdf?1664145632=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs_R_and_Kazh.pdf\u0026Expires=1732762200\u0026Signature=RKyJJRTusqDxYjP9FRte-IpTOvXKUN50MCIsgKDRCm~rBMn4uLa0NW4WnbAt6LbYtfQdiM2RuBDXLABq3KHdchIEFGZJt0GbKKokYBKDaVAA7DMaTyVDtLR-Q6bwJtWWy3FZPCrS-yCo0duEikqFwTTPQGnhmDd3Y7v8y6d6phE6L8igVm0D4xX5p-LT9iJTXZHHF5qDtFH8TFxuw9YE-ua9fUUXqXSiMYlJqpVNS~cEsu1vFdPoNGuCwM~azy8RJzM~Ppo9bktTXM3T8MGbbdtamLmvBgtcNu6PYNvUqCLoHrkpH1~o3VF~3jtFS3h9fekxFf9LspPr16JzM3SDEw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials","translated_slug":"","page_count":16,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":91541857,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/91541857/thumbnails/1.jpg","file_name":"1211.4305v1.pdf","download_url":"https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs_R_and_Kazh.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/91541857/1211.4305v1-libre.pdf?1664145632=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs_R_and_Kazh.pdf\u0026Expires=1732762200\u0026Signature=RKyJJRTusqDxYjP9FRte-IpTOvXKUN50MCIsgKDRCm~rBMn4uLa0NW4WnbAt6LbYtfQdiM2RuBDXLABq3KHdchIEFGZJt0GbKKokYBKDaVAA7DMaTyVDtLR-Q6bwJtWWy3FZPCrS-yCo0duEikqFwTTPQGnhmDd3Y7v8y6d6phE6L8igVm0D4xX5p-LT9iJTXZHHF5qDtFH8TFxuw9YE-ua9fUUXqXSiMYlJqpVNS~cEsu1vFdPoNGuCwM~azy8RJzM~Ppo9bktTXM3T8MGbbdtamLmvBgtcNu6PYNvUqCLoHrkpH1~o3VF~3jtFS3h9fekxFf9LspPr16JzM3SDEw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="15897358" id="papers"><div class="js-work-strip profile--work_container" data-work-id="95559998"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559998/Schubert_Numbers"><img alt="Research paper thumbnail of Schubert Numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/97707786/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559998/Schubert_Numbers">Schubert Numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This thesis discusses intersections of the Schubert varieties in the flag variety associated to a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4caf13a14dae553799c5e3bcba915cdd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707786,"asset_id":95559998,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559998"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559998"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559998; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559998]").text(description); $(".js-view-count[data-work-id=95559998]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559998; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559998']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559998, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4caf13a14dae553799c5e3bcba915cdd" } } $('.js-work-strip[data-work-id=95559998]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559998,"title":"Schubert Numbers","translated_title":"","metadata":{"abstract":"This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders","publisher":"TRACE: Tennessee Research and Creative Exchange","publication_date":{"day":1,"month":5,"year":2010,"errors":{}}},"translated_abstract":"This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders","internal_url":"https://www.academia.edu/95559998/Schubert_Numbers","translated_internal_url":"","created_at":"2023-01-23T14:01:44.369-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707786,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707786/thumbnails/1.jpg","file_name":"268764458.pdf","download_url":"https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Schubert_Numbers.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707786/268764458-libre.pdf?1674512167=\u0026response-content-disposition=attachment%3B+filename%3DSchubert_Numbers.pdf\u0026Expires=1732762200\u0026Signature=LdU5SPOZb9S6vuw8lkb96BCciCz0zxkleedwf1a6AL5kcw53MDfKIgPKYJdjlmsaHK0tuSQnpp9bfTbMfmUyCUFCn8dRgUR7LJ6s3VR2x6OsLlL60BUzLgQ-2rYd0te068oKylbtYmU3AKqaFu9HrAcpFHq0CLNXqL2OuxhIbT0HH2u6YQaJSoIqb04wWWr0KyVW1aZBjaOK~fj3mwEous-babLLQ10MGLweS3k2dnWT8jx3mEhyzOaKxBWzFB49NEJAOWs~x3qsXMFcd96TkNqdVzFGJsvoTc6hjerGhS4WVWkzaZTk7~fSDh6y7-~TTwLmj4-oWc~zEQtnaWWjBQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Schubert_Numbers","translated_slug":"","page_count":71,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707786,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707786/thumbnails/1.jpg","file_name":"268764458.pdf","download_url":"https://www.academia.edu/attachments/97707786/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Schubert_Numbers.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707786/268764458-libre.pdf?1674512167=\u0026response-content-disposition=attachment%3B+filename%3DSchubert_Numbers.pdf\u0026Expires=1732762200\u0026Signature=LdU5SPOZb9S6vuw8lkb96BCciCz0zxkleedwf1a6AL5kcw53MDfKIgPKYJdjlmsaHK0tuSQnpp9bfTbMfmUyCUFCn8dRgUR7LJ6s3VR2x6OsLlL60BUzLgQ-2rYd0te068oKylbtYmU3AKqaFu9HrAcpFHq0CLNXqL2OuxhIbT0HH2u6YQaJSoIqb04wWWr0KyVW1aZBjaOK~fj3mwEous-babLLQ10MGLweS3k2dnWT8jx3mEhyzOaKxBWzFB49NEJAOWs~x3qsXMFcd96TkNqdVzFGJsvoTc6hjerGhS4WVWkzaZTk7~fSDh6y7-~TTwLmj4-oWc~zEQtnaWWjBQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330395,"url":"https://core.ac.uk/download/268764458.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559996"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs"><img alt="Research paper thumbnail of Inequalities on Bruhat graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/97707819/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs">Inequalities on Bruhat graphs</a></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c2f89d98ce649fa6e9d0c89ccade0d67" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707819,"asset_id":95559996,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559996"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559996"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559996; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559996]").text(description); $(".js-view-count[data-work-id=95559996]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559996; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559996']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559996, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c2f89d98ce649fa6e9d0c89ccade0d67" } } $('.js-work-strip[data-work-id=95559996]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559996,"title":"Inequalities on Bruhat graphs","translated_title":"","metadata":{"grobid_abstract":"From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"grobid_abstract_attachment_id":97707819},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559996/Inequalities_on_Bruhat_graphs","translated_internal_url":"","created_at":"2023-01-23T14:01:44.220-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707819,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707819/thumbnails/1.jpg","file_name":"1211.4305.pdf","download_url":"https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707819/1211.4305-libre.pdf?1674512140=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs.pdf\u0026Expires=1732762200\u0026Signature=cTr-4pT0rAqk462~XALwXTQoSArnl85ZqRl~vmI3vvtj8KO9UR4iqmKuu9InV4OMCr31KtG1InWxMg~h4w5kRRsAJf2Nzz4ZUO4auu7ldLrmzRPogtrOqMJSIokUpzaZvd-4BnCkuc~ggOSpNDw1EANzvI7W~jDCrl5nFYT36GdrFppgF7LD9nE7fD8Cs-2MOCo31OObc4A7HhmiXYKSeyqovDyedyIdYVk3Ub0xP9QgATf2hzR4XKMCa6ZBbG4iw97ylm3X4ctFlBC6eu~EC~i13F2ghjLWSi79DKzLG7OauNX9N8OqMOr~OANwlYKnKtpVLHg83XjFht1gC5a2-w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Inequalities_on_Bruhat_graphs","translated_slug":"","page_count":16,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707819,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707819/thumbnails/1.jpg","file_name":"1211.4305.pdf","download_url":"https://www.academia.edu/attachments/97707819/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707819/1211.4305-libre.pdf?1674512140=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs.pdf\u0026Expires=1732762200\u0026Signature=cTr-4pT0rAqk462~XALwXTQoSArnl85ZqRl~vmI3vvtj8KO9UR4iqmKuu9InV4OMCr31KtG1InWxMg~h4w5kRRsAJf2Nzz4ZUO4auu7ldLrmzRPogtrOqMJSIokUpzaZvd-4BnCkuc~ggOSpNDw1EANzvI7W~jDCrl5nFYT36GdrFppgF7LD9nE7fD8Cs-2MOCo31OObc4A7HhmiXYKSeyqovDyedyIdYVk3Ub0xP9QgATf2hzR4XKMCa6ZBbG4iw97ylm3X4ctFlBC6eu~EC~i13F2ghjLWSi79DKzLG7OauNX9N8OqMOr~OANwlYKnKtpVLHg83XjFht1gC5a2-w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330394,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.3828\u0026rep=rep1\u0026type=pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559993"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups"><img alt="Research paper thumbnail of Bruhat order and graph structures of lower intervals in Coxeter groups" class="work-thumbnail" src="https://attachments.academia-assets.com/97707783/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups">Bruhat order and graph structures of lower intervals in Coxeter groups</a></div><div class="wp-workCard_item"><span>arXiv: Combinatorics</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-side...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="bc2b6a3236c97d82c35bbbd30522bf5b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707783,"asset_id":95559993,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559993"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559993"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559993; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559993]").text(description); $(".js-view-count[data-work-id=95559993]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559993; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559993']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559993, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "bc2b6a3236c97d82c35bbbd30522bf5b" } } $('.js-work-strip[data-work-id=95559993]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559993,"title":"Bruhat order and graph structures of lower intervals in Coxeter groups","translated_title":"","metadata":{"abstract":"We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.","publication_date":{"day":null,"month":null,"year":2018,"errors":{}},"publication_name":"arXiv: Combinatorics"},"translated_abstract":"We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.","internal_url":"https://www.academia.edu/95559993/Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups","translated_internal_url":"","created_at":"2023-01-23T14:01:44.085-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707783,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707783/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707783/1812.06266v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=WkMOBcSxj54WMQCptk1Wn0Z7A3UAORij8QRjW84m12aKizSGaYkasSzfZVLp6JXPRdNuE~AojVNvS9MGPrLTAMZSOkyCKIUOaLXQdBjGS6ebnN6Gw9h6xyFiEZ8DKOJF2TP5Wg-BpgRuuQrVQp3w7Aj3fEOM5GLefl4XIs4tOl1UKzHznh36rs-c2DQhBBj1tr-8BjkhaydNlCbpDhH-6H3x549En4HxhKdWGjoMDw4vIJIHBt9bAKKPpnHDm2i3xmrJqLWWXyH6WCF~UhSJs0hV-w5JH~lJibnykpX5S~jRm7O4pQdeYAdZmkSwkIgABnhPGBFwJr54yqLSGFaMCw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Bruhat_order_and_graph_structures_of_lower_intervals_in_Coxeter_groups","translated_slug":"","page_count":18,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707783,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707783/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707783/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707783/1812.06266v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=WkMOBcSxj54WMQCptk1Wn0Z7A3UAORij8QRjW84m12aKizSGaYkasSzfZVLp6JXPRdNuE~AojVNvS9MGPrLTAMZSOkyCKIUOaLXQdBjGS6ebnN6Gw9h6xyFiEZ8DKOJF2TP5Wg-BpgRuuQrVQp3w7Aj3fEOM5GLefl4XIs4tOl1UKzHznh36rs-c2DQhBBj1tr-8BjkhaydNlCbpDhH-6H3x549En4HxhKdWGjoMDw4vIJIHBt9bAKKPpnHDm2i3xmrJqLWWXyH6WCF~UhSJs0hV-w5JH~lJibnykpX5S~jRm7O4pQdeYAdZmkSwkIgABnhPGBFwJr54yqLSGFaMCw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707782,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707782/thumbnails/1.jpg","file_name":"1812.06266v1.pdf","download_url":"https://www.academia.edu/attachments/97707782/download_file","bulk_download_file_name":"Bruhat_order_and_graph_structures_of_low.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707782/1812.06266v1-libre.pdf?1674512147=\u0026response-content-disposition=attachment%3B+filename%3DBruhat_order_and_graph_structures_of_low.pdf\u0026Expires=1732762200\u0026Signature=S-Wb0525EoJPDaqmNb73koFrqvQfjvBWoi3Upp6Lhp~TBBfJTwHF5UWMAwccItt2UWwB7kLrICsFj967WNCxtUe0AZsw5UPx5unw3mCYZCza1UlI1FQq6cZX5RyifxlQ58gWHx2hVQHz-FcpaFzLenkTKehM2AgtxFB6CETpQmqr07Te7NlZxxI9~bUtqYxNzc8DmDSLGxY8J5MDrQjQ4twSf8HFh1k30UyC1O703Wq5-XpKS2~RhhjYL2p-e5yfI-SRnjZ2BmsaAAxiv~CiPicfxXdA1RzFF5oDp0v~Ri~HYl4L6Fv7b7dnk9BFcM9MFwfe8r7jtGWf-lKmJsDrsA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":2696636,"name":"quotient","url":"https://www.academia.edu/Documents/in/quotient"}],"urls":[{"id":28330392,"url":"https://arxiv.org/pdf/1812.06266v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559991"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials"><img alt="Research paper thumbnail of On Determinants and Signed Bigrassmannian Polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/97707784/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials">On Determinants and Signed Bigrassmannian Polynomials</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9ba5c6680ed617b5e3c32589bababc6f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707784,"asset_id":95559991,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559991"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559991"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559991; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559991]").text(description); $(".js-view-count[data-work-id=95559991]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559991; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559991']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559991, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9ba5c6680ed617b5e3c32589bababc6f" } } $('.js-work-strip[data-work-id=95559991]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559991,"title":"On Determinants and Signed Bigrassmannian Polynomials","translated_title":"","metadata":{"abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).","publication_date":{"day":null,"month":null,"year":2014,"errors":{}}},"translated_abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).","internal_url":"https://www.academia.edu/95559991/On_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_internal_url":"","created_at":"2023-01-23T14:01:43.954-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707784,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707784/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707784/_10_kobayashi-libre.pdf?1674512145=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=HeSV3mHlsAYCQkVd8H2dJezg8DVDD3hAJcGZ7PpdJ2WKRJ1CPL508CS90dEe8Ddmuw4dyZ3w0My0slb1~lat-YGpbEriaMGy2qC3N9Yy3oIJ6Le5ZHWHBpe~B4xHgkOn1PTAKx4cghOx3oe32F0BMc19x3F-ufsX6kRvFcADkEZ1iilg3v2h-UuwCzo8~xTg1btAGGIYMJgxAc3ueIuryHTSkJZQcFaUNcoqnVtEuBl4PM7YN9lCQOCbANQOrqNm5TzvdcJRO9MyzB0mRnFWjuKnukzaIR~RdAedpzLHxJMSvom0je-DgB0lBj3SX5puv0pwi-uPelGL15j5GLfoCg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_slug":"","page_count":14,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707784,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707784/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707784/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707784/_10_kobayashi-libre.pdf?1674512145=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=HeSV3mHlsAYCQkVd8H2dJezg8DVDD3hAJcGZ7PpdJ2WKRJ1CPL508CS90dEe8Ddmuw4dyZ3w0My0slb1~lat-YGpbEriaMGy2qC3N9Yy3oIJ6Le5ZHWHBpe~B4xHgkOn1PTAKx4cghOx3oe32F0BMc19x3F-ufsX6kRvFcADkEZ1iilg3v2h-UuwCzo8~xTg1btAGGIYMJgxAc3ueIuryHTSkJZQcFaUNcoqnVtEuBl4PM7YN9lCQOCbANQOrqNm5TzvdcJRO9MyzB0mRnFWjuKnukzaIR~RdAedpzLHxJMSvom0je-DgB0lBj3SX5puv0pwi-uPelGL15j5GLfoCg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707785,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707785/thumbnails/1.jpg","file_name":"_10_kobayashi.pdf","download_url":"https://www.academia.edu/attachments/97707785/download_file","bulk_download_file_name":"On_Determinants_and_Signed_Bigrassmannia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707785/_10_kobayashi-libre.pdf?1674512152=\u0026response-content-disposition=attachment%3B+filename%3DOn_Determinants_and_Signed_Bigrassmannia.pdf\u0026Expires=1732762200\u0026Signature=BNOb3~oppVvMhJ4Xw-z~apSiS7eqHRViYjf6C7xz3U6QFASpT55sDiCr9SsdhtrTk7XoWBXNCyERh45eTeQ6bg6cAbl7epUCi5gPPaNLRWMO2iuqiB5wRzLD93oBrZvdWb3rTG1hXqV-ecqpnbpzOx~lvYec-pe7FAxL4giEDOgABtYt07B70SAQG16jOVTbHvtWWnkD3aKCvnEEKdhGUJFZe9cW1FPYiQb6KBh-N7AQwGXwOECTn3cNfWyUrjEEk-gg7811THLPDZVNX1qNHwDKz0uPjUs9L22RxGybXQv4VDk9i~LD5UrFvLmGj-fj9Kju7tWTrRRepmLH7EGBxw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330391,"url":"http://www.math.okayama-u.ac.jp/mjou/mjou57/_10_kobayashi.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559989"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups"><img alt="Research paper thumbnail of Two-sided structure of double cosets in Coxeter groups" class="work-thumbnail" src="https://attachments.academia-assets.com/97707780/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups">Two-sided structure of double cosets in Coxeter groups</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this article is to show one clear difference between single (left or right) cosets and...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c4e079e53e104e137f77f2923e71f955" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707780,"asset_id":95559989,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559989"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559989"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559989; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559989]").text(description); $(".js-view-count[data-work-id=95559989]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559989; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559989']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559989, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c4e079e53e104e137f77f2923e71f955" } } $('.js-work-strip[data-work-id=95559989]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559989,"title":"Two-sided structure of double cosets in Coxeter groups","translated_title":"","metadata":{"abstract":"The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.","publication_date":{"day":null,"month":null,"year":2011,"errors":{}}},"translated_abstract":"The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.","internal_url":"https://www.academia.edu/95559989/Two_sided_structure_of_double_cosets_in_Coxeter_groups","translated_internal_url":"","created_at":"2023-01-23T14:01:43.834-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707780,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707780/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707780/162-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=EJb2PkQhcyxxktkNYp2uszsVq0AgWR1nvaGe5pJbBsrlljLWcYNecIJoH1midQ86USCWmKVGtplG24sjW5s~t7GDgCLzbkwGBGn6IDeZ23CDPJvGliHl-rGy0oluPBVMwJbL1pxE~ehx2Nxzsg6q1sr9sLeg8zHRKwKs2xTRslDbi76sau9lhr4yrx7B25K6h2RQwXfoOW4mCU9uLVvEcF~-6kuvT7Hsd9DlsPIqV8kLLdsmZ85EcnqIwS2M~R343W0cqOLtBSc0ReUFmpgZzuslE6ZP0tPZduRd-cuSJzBznYpPtVdu9XAeTZK-hgXi0K5DakR07uuQJdLk-OnnRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Two_sided_structure_of_double_cosets_in_Coxeter_groups","translated_slug":"","page_count":9,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707780,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707780/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707780/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707780/162-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=EJb2PkQhcyxxktkNYp2uszsVq0AgWR1nvaGe5pJbBsrlljLWcYNecIJoH1midQ86USCWmKVGtplG24sjW5s~t7GDgCLzbkwGBGn6IDeZ23CDPJvGliHl-rGy0oluPBVMwJbL1pxE~ehx2Nxzsg6q1sr9sLeg8zHRKwKs2xTRslDbi76sau9lhr4yrx7B25K6h2RQwXfoOW4mCU9uLVvEcF~-6kuvT7Hsd9DlsPIqV8kLLdsmZ85EcnqIwS2M~R343W0cqOLtBSc0ReUFmpgZzuslE6ZP0tPZduRd-cuSJzBznYpPtVdu9XAeTZK-hgXi0K5DakR07uuQJdLk-OnnRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707781,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707781/thumbnails/1.jpg","file_name":"162.pdf","download_url":"https://www.academia.edu/attachments/97707781/download_file","bulk_download_file_name":"Two_sided_structure_of_double_cosets_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707781/162-libre.pdf?1674512149=\u0026response-content-disposition=attachment%3B+filename%3DTwo_sided_structure_of_double_cosets_in.pdf\u0026Expires=1732762200\u0026Signature=OIL7F9rIvOrjTTh3ADTFmsDRJDXyEBvcKGiWwsL0Bm4Zr-8f~ia7NcaGs98Z-1mFlJ9hOeODXi4h6jmAJhQzD5ktN6tqj~N28cSX3LufniLPqylDHvDNznhakXQfV~qU643LB-uibDvxImanzZLyI4adXfsboa-XnD-OqBZEeewlbRNFg2heFkrceIGv14ecwRbPphchggype6H38Fm0o9NnVeYTOI6vGk6USiOkyTlVr8CMYWil4rkgHUr4pawKnP5M7XaqyDP40Ji8UTU5YkVi07iO81f~Xm4qUGiYl4EWnppJ7TPm1DRW-3oJaiimw4-7Epe2qjmTJLCCn78pxw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":28330390,"url":"http://www.math.titech.ac.jp/~tosho/Preprints/pdf/162.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559987"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set"><img alt="Research paper thumbnail of Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set" class="work-thumbnail" src="https://attachments.academia-assets.com/97707778/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set">Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set</a></div><div class="wp-workCard_item"><span>The Electronic Journal of Combinatorics</span><span>, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Bigrassmannian permutations are known as permutations which have precisely one left descent and o...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c6cafc084e5d32a4c7b83927867ebd68" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707778,"asset_id":95559987,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559987"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559987"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559987; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559987]").text(description); $(".js-view-count[data-work-id=95559987]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559987; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559987']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559987, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c6cafc084e5d32a4c7b83927867ebd68" } } $('.js-work-strip[data-work-id=95559987]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559987,"title":"Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set","translated_title":"","metadata":{"abstract":"Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","publisher":"The Electronic Journal of Combinatorics","publication_date":{"day":null,"month":null,"year":2010,"errors":{}},"publication_name":"The Electronic Journal of Combinatorics"},"translated_abstract":"Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","internal_url":"https://www.academia.edu/95559987/Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set","translated_internal_url":"","created_at":"2023-01-23T14:01:43.698-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707778,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707778/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707778/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=Gj13uQ8UsC-7q60wxTBbjSr8r4wlE5rTeFxPm8bnbxaUtaTobhI~FmqEAfTk-NWPj3TcLxWN66ZEBtFke9a-krNYtU-SUqvaDBl-5pjjnrRx-rj8SUhRjslSOEaja2oZE0vWJfT6FN7syM7tQYhkyX7wI~j9FV~pMOU5tmhmHpyespLgqMpQIrGU188feCp0K0ERn3yDDtusfY~AyN80GybWHLobdQDm~KmbkM0qihkHFuVW0SCt8il4AjccTRKgywQ3v3dtkA2ioIXT7~MWBNJ5yhLjYPFKSHkyEzai2DPf~N36fp~FEAOCXll-Npj~9hJrb5gI~ovvpQbViM4SJA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Bijection_Between_Bigrassmannian_Permutations_Maximal_below_a_Permutation_and_its_Essential_Set","translated_slug":"","page_count":8,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707778,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707778/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707778/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707778/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=Gj13uQ8UsC-7q60wxTBbjSr8r4wlE5rTeFxPm8bnbxaUtaTobhI~FmqEAfTk-NWPj3TcLxWN66ZEBtFke9a-krNYtU-SUqvaDBl-5pjjnrRx-rj8SUhRjslSOEaja2oZE0vWJfT6FN7syM7tQYhkyX7wI~j9FV~pMOU5tmhmHpyespLgqMpQIrGU188feCp0K0ERn3yDDtusfY~AyN80GybWHLobdQDm~KmbkM0qihkHFuVW0SCt8il4AjccTRKgywQ3v3dtkA2ioIXT7~MWBNJ5yhLjYPFKSHkyEzai2DPf~N36fp~FEAOCXll-Npj~9hJrb5gI~ovvpQbViM4SJA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707779,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707779/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/97707779/download_file","bulk_download_file_name":"Bijection_Between_Bigrassmannian_Permuta.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707779/pdf-libre.pdf?1674512144=\u0026response-content-disposition=attachment%3B+filename%3DBijection_Between_Bigrassmannian_Permuta.pdf\u0026Expires=1732762200\u0026Signature=eGq0L9vMfXOD6CcReTDiK3FSYAq5uXAmZuRxzObf3HpDH2Jvuh0KtdMwVlx5XULV~~eFZscwqYawODf9dMPUTdybprUr45OBx9CpzAFxaPbj19KoYT6uQ3XwFDLG62Su-JY01XVrIP6waNfjobNn26~BNcvzaH6XjyQ3xuhZe5QocaXuyO2gNIz23COBhnMryf3CDs-W~q-~tl7WP60ZncisL4b8BeQ5b2dyzVHD1hiZOkQKZpx0~auJnUgKs2fpTlKSm22brHLQZ4gklL7RnFg-BtjX9RR8KX-kBD0YMknyGHBY3Gulua7ePwTYrd4HthIc3TkGg24aD4kZcoI9Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":3381241,"name":"bijection","url":"https://www.academia.edu/Documents/in/bijection"}],"urls":[{"id":28330389,"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1n27/pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559983"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices"><img alt="Research paper thumbnail of Weighted Counting of Inversions on Alternating Sign Matrices" class="work-thumbnail" src="https://attachments.academia-assets.com/97707840/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices">Weighted Counting of Inversions on Alternating Sign Matrices</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2019</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fb494141832b24b0936ad068f166acd5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707840,"asset_id":95559983,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559983"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559983"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559983; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559983]").text(description); $(".js-view-count[data-work-id=95559983]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559983; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559983']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559983, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fb494141832b24b0936ad068f166acd5" } } $('.js-work-strip[data-work-id=95559983]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559983,"title":"Weighted Counting of Inversions on Alternating Sign Matrices","translated_title":"","metadata":{"publisher":"Springer Science and Business Media LLC","grobid_abstract":"We generalize the author's formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).","publication_date":{"day":null,"month":null,"year":2019,"errors":{}},"publication_name":"Order","grobid_abstract_attachment_id":97707840},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559983/Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices","translated_internal_url":"","created_at":"2023-01-23T14:01:43.385-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707840,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707840/thumbnails/1.jpg","file_name":"1904.pdf","download_url":"https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Weighted_Counting_of_Inversions_on_Alter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707840/1904-libre.pdf?1674512137=\u0026response-content-disposition=attachment%3B+filename%3DWeighted_Counting_of_Inversions_on_Alter.pdf\u0026Expires=1732762200\u0026Signature=YGf95hlhquu0XSk5NVSqTftx6-NRtb5BiY2K3f-beopX713mG8nLW6~AekMwEQOgYobGaTZBVYairqoTvuNHo7jnd-HOVSw1glPPS9E~RhSdOKIbXY-kg42cDwb4GbUV1h219vhrJAVqS76dqk1ykaY5bAh-4FGiVPKb7BccZCgTWfHhB0iIuEIiuiju3aY2CYtsc2RguooIViJa2eaGp6AU2mnl652fWNWVq6qGd0pL1uVxP1LCTtC-zZw7kU~itUPmGngPiyQbmcKXpOABp3YusnjaxgaW9W6vOjRVZZoZ20pG9unJYUtATp-CXI7ccnlGjjPaKGDdPdvFsRdnCA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Weighted_Counting_of_Inversions_on_Alternating_Sign_Matrices","translated_slug":"","page_count":19,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707840,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707840/thumbnails/1.jpg","file_name":"1904.pdf","download_url":"https://www.academia.edu/attachments/97707840/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Weighted_Counting_of_Inversions_on_Alter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707840/1904-libre.pdf?1674512137=\u0026response-content-disposition=attachment%3B+filename%3DWeighted_Counting_of_Inversions_on_Alter.pdf\u0026Expires=1732762200\u0026Signature=YGf95hlhquu0XSk5NVSqTftx6-NRtb5BiY2K3f-beopX713mG8nLW6~AekMwEQOgYobGaTZBVYairqoTvuNHo7jnd-HOVSw1glPPS9E~RhSdOKIbXY-kg42cDwb4GbUV1h219vhrJAVqS76dqk1ykaY5bAh-4FGiVPKb7BccZCgTWfHhB0iIuEIiuiju3aY2CYtsc2RguooIViJa2eaGp6AU2mnl652fWNWVq6qGd0pL1uVxP1LCTtC-zZw7kU~itUPmGngPiyQbmcKXpOABp3YusnjaxgaW9W6vOjRVZZoZ20pG9unJYUtATp-CXI7ccnlGjjPaKGDdPdvFsRdnCA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":183513,"name":"Order","url":"https://www.academia.edu/Documents/in/Order"}],"urls":[{"id":28330387,"url":"http://link.springer.com/content/pdf/10.1007/s11083-019-09515-1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices"><img alt="Research paper thumbnail of A directed graph structure of alternating sign matrices" class="work-thumbnail" src="https://attachments.academia-assets.com/97707825/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices">A directed graph structure of alternating sign matrices</a></div><div class="wp-workCard_item"><span>Linear Algebra and its Applications</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="72567e8df1cd7bfefc341deae85b8654" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707825,"asset_id":95559981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559981; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559981]").text(description); $(".js-view-count[data-work-id=95559981]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559981; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559981']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559981, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "72567e8df1cd7bfefc341deae85b8654" } } $('.js-work-strip[data-work-id=95559981]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559981,"title":"A directed graph structure of alternating sign matrices","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, q-analogs (which we name qTNN and qSFL properties). As a by-product, we obtain a new kind of permutation statistic, the signed bigrassmannian statistics, using Dodgson's condensation on determinants.","publication_date":{"day":null,"month":null,"year":2017,"errors":{}},"publication_name":"Linear Algebra and its Applications","grobid_abstract_attachment_id":97707825},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559981/A_directed_graph_structure_of_alternating_sign_matrices","translated_internal_url":"","created_at":"2023-01-23T14:01:43.260-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707825,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707825/thumbnails/1.jpg","file_name":"1903.pdf","download_url":"https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_directed_graph_structure_of_alternatin.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707825/1903-libre.pdf?1674512155=\u0026response-content-disposition=attachment%3B+filename%3DA_directed_graph_structure_of_alternatin.pdf\u0026Expires=1732762200\u0026Signature=YfFamCvJzSe5a-sBQf0aHK5gKsLOldDaITypzFgbT4RKTmd6T0R1TZ5nBVcNu3Xd5h4IYMUcyeNUMp~EUHHYXEcMeJJIPkHGtEx59v5gNKYO80jLq9jNQyGgtqe1vJOnAHUSDkj1MvXfzc7dQWTqbG8-sAqPBtGr0sfpFHVYfjUFf-P9Ge2ypCEvcNjgnhtNo4Ay7DIZzUAYDA14~g48AqKj3ULOlTUZe15rZg3PxdE10Ng-hUT1hYL7gD3HzgAwEWLUcHdrzpn2nA7kqfCpo3iSXkR6pzHTGvXv0dg1hZXtpXeaSL02I9VxJw2729Ooz0N9uTy~HQvNrJWVz3exSg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_directed_graph_structure_of_alternating_sign_matrices","translated_slug":"","page_count":27,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707825,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707825/thumbnails/1.jpg","file_name":"1903.pdf","download_url":"https://www.academia.edu/attachments/97707825/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_directed_graph_structure_of_alternatin.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707825/1903-libre.pdf?1674512155=\u0026response-content-disposition=attachment%3B+filename%3DA_directed_graph_structure_of_alternatin.pdf\u0026Expires=1732762200\u0026Signature=YfFamCvJzSe5a-sBQf0aHK5gKsLOldDaITypzFgbT4RKTmd6T0R1TZ5nBVcNu3Xd5h4IYMUcyeNUMp~EUHHYXEcMeJJIPkHGtEx59v5gNKYO80jLq9jNQyGgtqe1vJOnAHUSDkj1MvXfzc7dQWTqbG8-sAqPBtGr0sfpFHVYfjUFf-P9Ge2ypCEvcNjgnhtNo4Ay7DIZzUAYDA14~g48AqKj3ULOlTUZe15rZg3PxdE10Ng-hUT1hYL7gD3HzgAwEWLUcHdrzpn2nA7kqfCpo3iSXkR6pzHTGvXv0dg1hZXtpXeaSL02I9VxJw2729Ooz0N9uTy~HQvNrJWVz3exSg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":48,"name":"Engineering","url":"https://www.academia.edu/Documents/in/Engineering"},{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559977"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set"><img alt="Research paper thumbnail of More combinatorics of Fulton's essential set" class="work-thumbnail" src="https://attachments.academia-assets.com/97707824/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set">More combinatorics of Fulton's essential set</a></div><div class="wp-workCard_item"><span>International Mathematical Forum</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c9a48d45c133a9e6c19335f92ae70c4c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707824,"asset_id":95559977,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559977"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559977"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559977; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559977]").text(description); $(".js-view-count[data-work-id=95559977]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559977; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559977']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559977, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c9a48d45c133a9e6c19335f92ae70c4c" } } $('.js-work-strip[data-work-id=95559977]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559977,"title":"More combinatorics of Fulton's essential set","translated_title":"","metadata":{"publisher":"Hikari, Ltd.","ai_title_tag":"Combinatorics of Fulton's Essential Set and Baxter Permutations","grobid_abstract":"We develop combinatorics of Fulton's essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret Eriksson-Linusson's characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of Bruhat-Ehresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"International Mathematical Forum","grobid_abstract_attachment_id":97707824},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559977/More_combinatorics_of_Fultons_essential_set","translated_internal_url":"","created_at":"2023-01-23T14:01:42.895-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707824,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707824/thumbnails/1.jpg","file_name":"25baf70f8f7631bf25e03f00e2ab7ed7b5ec.pdf","download_url":"https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"More_combinatorics_of_Fultons_essential.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707824/25baf70f8f7631bf25e03f00e2ab7ed7b5ec-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DMore_combinatorics_of_Fultons_essential.pdf\u0026Expires=1732762200\u0026Signature=SDplmj9Y8UH5KXMgl3tQeXuLyu-YCoZ4HCJZaJbLea~JmRoahXlPzysKhqXTi1Pgd7jQVnCWUR-WwCrqNxCPOM4s205-ZCIhkfsHtQ73qRhu9nyHgrzKWxTv0oCUvq4l-MSvAZH5wCM-EQwEHs2qBCM47DkaOupk7w70S-bVhGKRS5RuBvzua6AM9pc-spv4GuUuS3jeOgaME8N6yqfNqwOSLBRxU4HHKUMVUbqbFQaGh6b4RlgjfXu4uZV3JwluqN-iEjm4eswThV7uCfg7SUErON7mksXN0-O8EHeiwksFHK2ktw3JIPt6ts8LyGaMnnSMVGpNSSKPLBaXgk2sYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"More_combinatorics_of_Fultons_essential_set","translated_slug":"","page_count":26,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707824,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707824/thumbnails/1.jpg","file_name":"25baf70f8f7631bf25e03f00e2ab7ed7b5ec.pdf","download_url":"https://www.academia.edu/attachments/97707824/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"More_combinatorics_of_Fultons_essential.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707824/25baf70f8f7631bf25e03f00e2ab7ed7b5ec-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DMore_combinatorics_of_Fultons_essential.pdf\u0026Expires=1732762200\u0026Signature=SDplmj9Y8UH5KXMgl3tQeXuLyu-YCoZ4HCJZaJbLea~JmRoahXlPzysKhqXTi1Pgd7jQVnCWUR-WwCrqNxCPOM4s205-ZCIhkfsHtQ73qRhu9nyHgrzKWxTv0oCUvq4l-MSvAZH5wCM-EQwEHs2qBCM47DkaOupk7w70S-bVhGKRS5RuBvzua6AM9pc-spv4GuUuS3jeOgaME8N6yqfNqwOSLBRxU4HHKUMVUbqbFQaGh6b4RlgjfXu4uZV3JwluqN-iEjm4eswThV7uCfg7SUErON7mksXN0-O8EHeiwksFHK2ktw3JIPt6ts8LyGaMnnSMVGpNSSKPLBaXgk2sYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":14193,"name":"Philosophy of Property","url":"https://www.academia.edu/Documents/in/Philosophy_of_Property"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559973"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order"><img alt="Research paper thumbnail of Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order" class="work-thumbnail" src="https://attachments.academia-assets.com/97707814/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order">Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2010</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f0dbf1f44016e58ec97666d95af6bd29" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707814,"asset_id":95559973,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559973"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559973"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559973; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559973]").text(description); $(".js-view-count[data-work-id=95559973]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559973; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559973']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559973, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f0dbf1f44016e58ec97666d95af6bd29" } } $('.js-work-strip[data-work-id=95559973]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559973,"title":"Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order","translated_title":"","metadata":{"publisher":"Springer Nature","grobid_abstract":"In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.","publication_date":{"day":null,"month":null,"year":2010,"errors":{}},"publication_name":"Order","grobid_abstract_attachment_id":97707814},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559973/Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order","translated_internal_url":"","created_at":"2023-01-23T14:01:42.638-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707814,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707814/thumbnails/1.jpg","file_name":"1005.pdf","download_url":"https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumeration_of_Bigrassmannian_Permutatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707814/1005-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DEnumeration_of_Bigrassmannian_Permutatio.pdf\u0026Expires=1732762200\u0026Signature=QSag-kw~VpYP0yfsg-9QfTpwkq9JNDgUBCeejAizmwoEhGfjpe-RlDJDzUWnnJ0Ito66b4Q6uIM81~Oz3rBjJXYOYA8~EkSYif-JBQyIPLVz0ctrmzu-30-COVijNUxJdRdswZk46p7RZwt-mb0QS47X73ZK3QCyBvZfAmusH5RtsxqJJhz~wJK~s0vtZjN8DRJ2ZEs1zHyO2znEajGDTvYtHllBnjwW7ScBTq5CCiOnfvSd8SpOvXGcvMhhxTcWLhuuGcTxYXBdiY08PVy56zYedS0dKfTFisHs5nppkLrWh34vDiUP34Ptt81~1-V7nKmldvTwe~XQ6RQblq4yFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Enumeration_of_Bigrassmannian_Permutations_Below_a_Permutation_in_Bruhat_Order","translated_slug":"","page_count":7,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707814,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707814/thumbnails/1.jpg","file_name":"1005.pdf","download_url":"https://www.academia.edu/attachments/97707814/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumeration_of_Bigrassmannian_Permutatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707814/1005-libre.pdf?1674512139=\u0026response-content-disposition=attachment%3B+filename%3DEnumeration_of_Bigrassmannian_Permutatio.pdf\u0026Expires=1732762200\u0026Signature=QSag-kw~VpYP0yfsg-9QfTpwkq9JNDgUBCeejAizmwoEhGfjpe-RlDJDzUWnnJ0Ito66b4Q6uIM81~Oz3rBjJXYOYA8~EkSYif-JBQyIPLVz0ctrmzu-30-COVijNUxJdRdswZk46p7RZwt-mb0QS47X73ZK3QCyBvZfAmusH5RtsxqJJhz~wJK~s0vtZjN8DRJ2ZEs1zHyO2znEajGDTvYtHllBnjwW7ScBTq5CCiOnfvSd8SpOvXGcvMhhxTcWLhuuGcTxYXBdiY08PVy56zYedS0dKfTFisHs5nppkLrWh34vDiUP34Ptt81~1-V7nKmldvTwe~XQ6RQblq4yFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":42156,"name":"Enumeration","url":"https://www.academia.edu/Documents/in/Enumeration"},{"id":150408,"name":"Symmetric group","url":"https://www.academia.edu/Documents/in/Symmetric_group"},{"id":183513,"name":"Order","url":"https://www.academia.edu/Documents/in/Order"},{"id":482747,"name":"Coxeter groups","url":"https://www.academia.edu/Documents/in/Coxeter_groups"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559942"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas"><img alt="Research paper thumbnail of Matrix representation of meet-irreducible discrete copulas" class="work-thumbnail" src="https://attachments.academia-assets.com/97707803/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas">Matrix representation of meet-irreducible discrete copulas</a></div><div class="wp-workCard_item"><span>Fuzzy Sets and Systems</span><span>, 2014</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="de59ac6c7d715973b371b7693cdf3d92" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707803,"asset_id":95559942,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559942"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559942"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559942; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559942]").text(description); $(".js-view-count[data-work-id=95559942]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559942; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559942']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559942, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "de59ac6c7d715973b371b7693cdf3d92" } } $('.js-work-strip[data-work-id=95559942]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559942,"title":"Matrix representation of meet-irreducible discrete copulas","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"Following Aguiló-Suñer-Torrens (2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we give characterizations of meet-irreducible discrete copulas from an order-theoretical aspect: we show that the set of all irreducible discrete copulas is a lattice in analogy with Nelsen andÚbeda-Flores (2005). Moreover, we clarify its lattice structure related to Kendall's τ and Spearman's ρ borrowing ideas from Coxeter groups.","publication_date":{"day":null,"month":null,"year":2014,"errors":{}},"publication_name":"Fuzzy Sets and Systems","grobid_abstract_attachment_id":97707803},"translated_abstract":null,"internal_url":"https://www.academia.edu/95559942/Matrix_representation_of_meet_irreducible_discrete_copulas","translated_internal_url":"","created_at":"2023-01-23T14:01:38.767-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707803,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707803/thumbnails/1.jpg","file_name":"2108.13564v1.pdf","download_url":"https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Matrix_representation_of_meet_irreducibl.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707803/2108.13564v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DMatrix_representation_of_meet_irreducibl.pdf\u0026Expires=1732762200\u0026Signature=eaUKAH45MRB1OEh2RgY5TwFR141DeSw3CEFuMMt2FstjsuFmQWz50~pUdYxtmPveHNMr-TrnBz4krhQzLN74kKjBuW90Hv5SpEe-YcFbMMzRzPd9s7Un~qSu8wMqS0rT1WoE-y2~Sp~y9VSLqJnFi2T2qHZAGADFMf9WlcgfECiU0M-o38QIeHbqXlS~tYYwEgBO1RI-3pWjSxXFizfMaT28orN2UWF4plnpVvMl5EhtiF3-Q1ebeR5sk4rql9N8rZY1Z3xyOgn~8G03YtGj1mogI~~Zg1cKIzgSBVVRyud3aexOHGwUR7Yty1HOFDaACKF0Ufsp4VAW8wlxGOHd5Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Matrix_representation_of_meet_irreducible_discrete_copulas","translated_slug":"","page_count":20,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707803,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707803/thumbnails/1.jpg","file_name":"2108.13564v1.pdf","download_url":"https://www.academia.edu/attachments/97707803/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Matrix_representation_of_meet_irreducibl.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707803/2108.13564v1-libre.pdf?1674512148=\u0026response-content-disposition=attachment%3B+filename%3DMatrix_representation_of_meet_irreducibl.pdf\u0026Expires=1732762200\u0026Signature=eaUKAH45MRB1OEh2RgY5TwFR141DeSw3CEFuMMt2FstjsuFmQWz50~pUdYxtmPveHNMr-TrnBz4krhQzLN74kKjBuW90Hv5SpEe-YcFbMMzRzPd9s7Un~qSu8wMqS0rT1WoE-y2~Sp~y9VSLqJnFi2T2qHZAGADFMf9WlcgfECiU0M-o38QIeHbqXlS~tYYwEgBO1RI-3pWjSxXFizfMaT28orN2UWF4plnpVvMl5EhtiF3-Q1ebeR5sk4rql9N8rZY1Z3xyOgn~8G03YtGj1mogI~~Zg1cKIzgSBVVRyud3aexOHGwUR7Yty1HOFDaACKF0Ufsp4VAW8wlxGOHd5Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":115333,"name":"Analogy","url":"https://www.academia.edu/Documents/in/Analogy"},{"id":652157,"name":"Fuzzy Sets and Systems","url":"https://www.academia.edu/Documents/in/Fuzzy_Sets_and_Systems"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="95559894"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials"><img alt="Research paper thumbnail of Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/97707721/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials">Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="65409176c16466f98994a8aded64f879" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97707721,"asset_id":95559894,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="95559894"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="95559894"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 95559894; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=95559894]").text(description); $(".js-view-count[data-work-id=95559894]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 95559894; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='95559894']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 95559894, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "65409176c16466f98994a8aded64f879" } } $('.js-work-strip[data-work-id=95559894]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":95559894,"title":"Enumerative Combinatorics on Determinants and Signed Bigrassmannian Polynomials","translated_title":"","metadata":{"abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)","publisher":"Department of Mathematics, Faculty of Science, Okayama University","publication_date":{"day":null,"month":null,"year":2015,"errors":{}}},"translated_abstract":"As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)","internal_url":"https://www.academia.edu/95559894/Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_internal_url":"","created_at":"2023-01-23T14:00:13.123-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":97707721,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707721/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707721/32591052-libre.pdf?1674512153=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=J0c~FzsZDwSpmKOTmjw0chFevWTxzYTztq3iXMThuwRqe1RyjwByIDdKZKh0E9YptE4xtgmZgLgzizXgAERr9fIHdH9Y2wCfbRIRVOXlvO6VPsRUj52-gbGtFTYLtoYUbojNIPbzKXlyYdpD1H7TaioDIxQgl8MUqkCqOpkth0TWxbZHknLm9-LH~vtIez223GEmk6HUTZqCnFuYylndrKIvDKB4YDa~84vavZ38-odUdyQYrexFUj5ikNVXRydWJo7WGGixD1IPBz~3cPWJOw10TIGbULVxcsEfQ-arqpcxy0eJDKzWfbO63F8b~WjFf0Kahfo9U9flpPd6sDR3Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Enumerative_Combinatorics_on_Determinants_and_Signed_Bigrassmannian_Polynomials","translated_slug":"","page_count":14,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":97707721,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707721/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707721/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707721/32591052-libre.pdf?1674512153=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=J0c~FzsZDwSpmKOTmjw0chFevWTxzYTztq3iXMThuwRqe1RyjwByIDdKZKh0E9YptE4xtgmZgLgzizXgAERr9fIHdH9Y2wCfbRIRVOXlvO6VPsRUj52-gbGtFTYLtoYUbojNIPbzKXlyYdpD1H7TaioDIxQgl8MUqkCqOpkth0TWxbZHknLm9-LH~vtIez223GEmk6HUTZqCnFuYylndrKIvDKB4YDa~84vavZ38-odUdyQYrexFUj5ikNVXRydWJo7WGGixD1IPBz~3cPWJOw10TIGbULVxcsEfQ-arqpcxy0eJDKzWfbO63F8b~WjFf0Kahfo9U9flpPd6sDR3Kg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":97707720,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97707720/thumbnails/1.jpg","file_name":"32591052.pdf","download_url":"https://www.academia.edu/attachments/97707720/download_file","bulk_download_file_name":"Enumerative_Combinatorics_on_Determinant.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97707720/32591052-libre.pdf?1674512151=\u0026response-content-disposition=attachment%3B+filename%3DEnumerative_Combinatorics_on_Determinant.pdf\u0026Expires=1732762200\u0026Signature=Fc~p8HnpBlie1beovhMVqhftdkNCHWZ5HW2VZtsS5Nks5Dn63EdYM-rQM-V92kzZY3c3kP41RynJtp~npHHLPlLP1mkLveta1DcOpl5HpbM76CrNB20sUDuEiB48bXXp7W4~5sK3mVjtExIvLNmwZXi4DJhSyWyZmCbRZoWSTkbdGioevpHGUR1HpjUcq2P2e-uzGjRCCwgPesJCi~QNPeW-r9onBtFT1bm5CemYLbqJtV2HGreZ79-mmT0PS6~GOFdfEI0p8H9O2j~fjntjOYyoXOaSSD2aTtdk4OyjBTtv0pEUTC3taZXFXUSW4uOJTJjaquW32P0QawmDy3Xsew__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":20428,"name":"Tournaments","url":"https://www.academia.edu/Documents/in/Tournaments"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":2508371,"name":"Macdonald polynomials","url":"https://www.academia.edu/Documents/in/Macdonald_polynomials"}],"urls":[{"id":28330345,"url":"https://core.ac.uk/download/32591052.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="87291472"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials"><img alt="Research paper thumbnail of Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/91541857/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials">Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials</a></div><div class="wp-workCard_item"><span>Journal of Combinatorial Theory, Series A</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4829426e06b1493a6c8129d03bad7447" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":91541857,"asset_id":87291472,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="87291472"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="87291472"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 87291472; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=87291472]").text(description); $(".js-view-count[data-work-id=87291472]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 87291472; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='87291472']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 87291472, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4829426e06b1493a6c8129d03bad7447" } } $('.js-work-strip[data-work-id=87291472]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":87291472,"title":"Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"Journal of Combinatorial Theory, Series A","grobid_abstract_attachment_id":91541857},"translated_abstract":null,"internal_url":"https://www.academia.edu/87291472/Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials","translated_internal_url":"","created_at":"2022-09-25T15:27:08.696-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":216023629,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":91541857,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/91541857/thumbnails/1.jpg","file_name":"1211.4305v1.pdf","download_url":"https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs_R_and_Kazh.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/91541857/1211.4305v1-libre.pdf?1664145632=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs_R_and_Kazh.pdf\u0026Expires=1732762200\u0026Signature=RKyJJRTusqDxYjP9FRte-IpTOvXKUN50MCIsgKDRCm~rBMn4uLa0NW4WnbAt6LbYtfQdiM2RuBDXLABq3KHdchIEFGZJt0GbKKokYBKDaVAA7DMaTyVDtLR-Q6bwJtWWy3FZPCrS-yCo0duEikqFwTTPQGnhmDd3Y7v8y6d6phE6L8igVm0D4xX5p-LT9iJTXZHHF5qDtFH8TFxuw9YE-ua9fUUXqXSiMYlJqpVNS~cEsu1vFdPoNGuCwM~azy8RJzM~Ppo9bktTXM3T8MGbbdtamLmvBgtcNu6PYNvUqCLoHrkpH1~o3VF~3jtFS3h9fekxFf9LspPr16JzM3SDEw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Inequalities_on_Bruhat_graphs_R_and_Kazhdan_Lusztig_polynomials","translated_slug":"","page_count":16,"language":"en","content_type":"Work","owner":{"id":216023629,"first_name":"Masato","middle_initials":null,"last_name":"Kobayashi","page_name":"MasatoKobayashi4","domain_name":"independent","created_at":"2022-02-27T15:57:03.270-08:00","display_name":"Masato Kobayashi","url":"https://independent.academia.edu/MasatoKobayashi4"},"attachments":[{"id":91541857,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/91541857/thumbnails/1.jpg","file_name":"1211.4305v1.pdf","download_url":"https://www.academia.edu/attachments/91541857/download_file?st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&st=MTczMjc1ODYwMCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Inequalities_on_Bruhat_graphs_R_and_Kazh.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/91541857/1211.4305v1-libre.pdf?1664145632=\u0026response-content-disposition=attachment%3B+filename%3DInequalities_on_Bruhat_graphs_R_and_Kazh.pdf\u0026Expires=1732762200\u0026Signature=RKyJJRTusqDxYjP9FRte-IpTOvXKUN50MCIsgKDRCm~rBMn4uLa0NW4WnbAt6LbYtfQdiM2RuBDXLABq3KHdchIEFGZJt0GbKKokYBKDaVAA7DMaTyVDtLR-Q6bwJtWWy3FZPCrS-yCo0duEikqFwTTPQGnhmDd3Y7v8y6d6phE6L8igVm0D4xX5p-LT9iJTXZHHF5qDtFH8TFxuw9YE-ua9fUUXqXSiMYlJqpVNS~cEsu1vFdPoNGuCwM~azy8RJzM~Ppo9bktTXM3T8MGbbdtamLmvBgtcNu6PYNvUqCLoHrkpH1~o3VF~3jtFS3h9fekxFf9LspPr16JzM3SDEw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/google_contacts-0dfb882d836b94dbcb4a2d123d6933fc9533eda5be911641f20b4eb428429600.js"], function() { // from javascript_helper.rb $('.js-google-connect-button').click(function(e) { e.preventDefault(); GoogleContacts.authorize_and_show_contacts(); Aedu.Dismissibles.recordClickthrough("WowProfileImportContactsPrompt"); }); $('.js-update-biography-button').click(function(e) { e.preventDefault(); Aedu.Dismissibles.recordClickthrough("UpdateUserBiographyPrompt"); $.ajax({ url: $r.api_v0_profiles_update_about_path({ subdomain_param: 'api', about: "", }), type: 'PUT', success: function(response) { location.reload(); } }); }); $('.js-work-creator-button').click(function (e) { e.preventDefault(); window.location = $r.upload_funnel_document_path({ source: encodeURIComponent(""), }); }); $('.js-video-upload-button').click(function (e) { e.preventDefault(); window.location = $r.upload_funnel_video_path({ source: encodeURIComponent(""), }); }); $('.js-do-this-later-button').click(function() { $(this).closest('.js-profile-nag-panel').remove(); Aedu.Dismissibles.recordDismissal("WowProfileImportContactsPrompt"); }); $('.js-update-biography-do-this-later-button').click(function(){ $(this).closest('.js-profile-nag-panel').remove(); Aedu.Dismissibles.recordDismissal("UpdateUserBiographyPrompt"); }); $('.wow-profile-mentions-upsell--close').click(function(){ $('.wow-profile-mentions-upsell--panel').hide(); Aedu.Dismissibles.recordDismissal("WowProfileMentionsUpsell"); }); $('.wow-profile-mentions-upsell--button').click(function(){ Aedu.Dismissibles.recordClickthrough("WowProfileMentionsUpsell"); }); new WowProfile.SocialRedesignUserWorks({ initialWorksOffset: 20, allWorksOffset: 20, maxSections: 1 }) }); </script> </div></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile_edit-5ea339ee107c863779f560dd7275595239fed73f1a13d279d2b599a28c0ecd33.js","https://a.academia-assets.com/assets/add_coauthor-22174b608f9cb871d03443cafa7feac496fb50d7df2d66a53f5ee3c04ba67f53.js","https://a.academia-assets.com/assets/tab-dcac0130902f0cc2d8cb403714dd47454f11fc6fb0e99ae6a0827b06613abc20.js","https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js"], function() { // from javascript_helper.rb window.ae = window.ae || {}; window.ae.WowProfile = window.ae.WowProfile || {}; if(Aedu.User.current && Aedu.User.current.id === $viewedUser.id) { window.ae.WowProfile.current_user_edit = {}; new WowProfileEdit.EditUploadView({ el: '.js-edit-upload-button-wrapper', model: window.$current_user, }); new AddCoauthor.AddCoauthorsController(); } var userInfoView = new WowProfile.SocialRedesignUserInfo({ recaptcha_key: "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB" }); WowProfile.router = new WowProfile.Router({ userInfoView: userInfoView }); Backbone.history.start({ pushState: true, root: "/" + $viewedUser.page_name }); new WowProfile.UserWorksNav() }); </script> </div> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span   ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "2c5924486603213a0aee98d861e6ccfc97bf171af2690ce8df19521f4eda31b6", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="Oa7jWwvuCkcvfpW1wpXbIaJe2bPZUQhpI1XEImfholYR52zG/2a9mGOmxPh14bANb9jXEqynxlPgXi7p3x5txA==" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://independent.academia.edu/MasatoKobayashi4" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="UbGquMi3hCajZmczLmYHvFoisbTVB1/mjKd0K7302YJ5+CUlPD8z+e++Nn6ZEmyQl6S/FaDxkdxPrJ7gBQsWEA==" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><script src="https://recaptcha.net/recaptcha/api.js" async defer></script> <script> var invisibleRecaptchaSubmit = function () { var closestForm = function (ele) { var curEle = ele.parentNode; while (curEle.nodeName !== 'FORM' && curEle.nodeName !== 'BODY'){ curEle = curEle.parentNode; } return curEle.nodeName === 'FORM' ? curEle : null }; var eles = document.getElementsByClassName('g-recaptcha'); if (eles.length > 0) { var form = closestForm(eles[0]); if (form) { form.submit(); } } }; </script> <input type="submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" data-callback="invisibleRecaptchaSubmit" class="g-recaptcha btn btn-primary btn-block" value="Email me a link" value=""/> </form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div class="bootstrap" id="footer"><div class="footer-content clearfix text-center padding-top-7x" style="width:100%;"><ul class="footer-links-secondary footer-links-wide list-inline margin-bottom-1x"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a rel="nofollow" href="https://medium.com/academia">Blog</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/journals">Academia.edu Journals</a></li><li><a rel="nofollow" href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a rel="nofollow" href="https://support.academia.edu/"><svg style="width: 12px; height: 12px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer-links-tertiary list-inline margin-bottom-1x"><li class="small">Find new research papers in:</li><li class="small"><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul></div></div><div class="DesignSystem" id="credit" style="width:100%;"><ul class="u-pl0x footer-links-legal list-inline"><li><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2024</li></ul></div><script> //<![CDATA[ window.detect_gmtoffset = true; window.Academia && window.Academia.set_gmtoffset && Academia.set_gmtoffset('/gmtoffset'); //]]> </script> <div id='overlay_background'></div> <div id='bootstrap-modal-container' class='bootstrap'></div> <div id='ds-modal-container' class='bootstrap DesignSystem'></div> <div id='full-screen-modal'></div> </div> </body> </html>