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The first identity relates them with Eulerian numbers and" /> <meta name="twitter:image" content="https://0.academia-photos.com/96933265/88175321/76867386/s200_david.garber.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/84933692/Some_Identities_Involving_Second_Kind_Stirling_Numbers_of_Types_B_and_D_" /> <meta property="og:title" content="Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Using Reiner&#39;s definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and" /> <meta property="article:author" content="https://hit-il.academia.edu/DavidGarber" /> <meta name="description" content="Using Reiner&#39;s definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. 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The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $\\mathbb{R}[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.","author":[{"@context":"https://schema.org","@type":"Person","name":"David Garber","url":"https://hit-il.academia.edu/DavidGarber"}],"contributor":[],"dateCreated":"2022-08-16","dateModified":"2025-02-02","datePublished":"2019-01-01","headline":"Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$","image":"https://attachments.academia-assets.com/89790571/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Computer Science","Pure Mathematics","Eulerian Numbers","Stirling engine","Coxeter groups","Hyperplane Arrangements"],"publication":"The Electronic Journal of Combinatorics","publisher":{"@context":"https://schema.org","@type":"Organization","name":"The Electronic Journal of 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{"work":{"id":84933692,"created_at":"2022-08-16T20:27:03.346-07:00","from_world_paper_id":212796516,"updated_at":"2025-02-02T12:23:30.942-08:00","_data":{"abstract":"Using Reiner\u0026#39;s definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $\\mathbb{R}[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.","publisher":"The Electronic Journal of Combinatorics","ai_title_tag":"Generalized Identities for Stirling Numbers","publication_date":"2019,,","publication_name":"The Electronic Journal of Combinatorics"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [96933265]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":89790571,"attachmentType":"pdf"}"><img alt="First page of “Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/89790571/mini_magick20220817-1-zcb0zp.png?1660706903" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Some Identities Involving Second Kind Stirling Numbers of Types $B$ and $D$</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="96933265" href="https://hit-il.academia.edu/DavidGarber"><img alt="Profile image of David Garber" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/96933265/88175321/76867386/s65_david.garber.jpeg" />David Garber</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2019, The Electronic Journal of Combinatorics</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">20 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 84933692; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Using Reiner&#39;s definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $\mathbb{R}[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":89790571,"attachmentType":"pdf","workUrl":"https://www.academia.edu/84933692/Some_Identities_Involving_Second_Kind_Stirling_Numbers_of_Types_B_and_D_"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":89790571,"attachmentType":"pdf","workUrl":"https://www.academia.edu/84933692/Some_Identities_Involving_Second_Kind_Stirling_Numbers_of_Types_B_and_D_"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="84933692" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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By using these equation, we derive some relations and identities related to these numbers and polynomials. Furthermore, by applying padic Volkenborn integral to these polynomials, we also derive some new identities for the generalized -Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials","attachmentId":77030806,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429686/Identities_Associated_with_Generalized_Stirling_Type_Numbers_and_Eulerian_Type_Polynomials","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65429686/Identities_Associated_with_Generalized_Stirling_Type_Numbers_and_Eulerian_Type_Polynomials"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="50223528" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50223528/Explicit_Formulas_and_Combinatorial_Identities_for_Generalized_Stirling_Numbers">Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3853163" href="https://mansoura.academia.edu/BeihElDesouky">Beih El-Desouky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, a modified approach to the multiparameter non-central Stirling numbers via differential operators, introduced by El-Desouky, and new explicit formulae of both kinds of these numbers are given. Also, some relations between these numbers and the generalized Hermite and Truesdel polynomials are obtained. Moreover, we investigate some new results for the generalized Stirling-type pair of Hsu and Shiue. Furthermore some interesting special cases, new combinatorial identities and a matrix representation are deduced.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers","attachmentId":68289772,"attachmentType":"pdf","work_url":"https://www.academia.edu/50223528/Explicit_Formulas_and_Combinatorial_Identities_for_Generalized_Stirling_Numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50223528/Explicit_Formulas_and_Combinatorial_Identities_for_Generalized_Stirling_Numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="50930553" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50930553/A_Family_of_Generalized_Stirling_Numbers_of_the_First_Kind">A Family of Generalized Stirling Numbers of the First Kind</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3853163" href="https://mansoura.academia.edu/BeihElDesouky">Beih El-Desouky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Applied Mathematics, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky [1] and Gould [2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Family of Generalized Stirling Numbers of the First Kind","attachmentId":68807318,"attachmentType":"pdf","work_url":"https://www.academia.edu/50930553/A_Family_of_Generalized_Stirling_Numbers_of_the_First_Kind","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50930553/A_Family_of_Generalized_Stirling_Numbers_of_the_First_Kind"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="95173158" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/95173158/Combinatorial_approach_of_certain_generalized_Stirling_numbers">Combinatorial approach of certain generalized Stirling numbers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="125884173" href="https://independent.academia.edu/HaceneBelbachir">Hacene Belbachir</a></div><p class="ds-related-work--abstract ds2-5-body-sm">A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Combinatorial approach of certain generalized Stirling numbers","attachmentId":97428152,"attachmentType":"pdf","work_url":"https://www.academia.edu/95173158/Combinatorial_approach_of_certain_generalized_Stirling_numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/95173158/Combinatorial_approach_of_certain_generalized_Stirling_numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="85625135" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/85625135/Certain_Identities_Involving_Stirling_Numbers_of_the_First_Kind">Certain Identities Involving Stirling Numbers of the First Kind</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39245771" href="https://independent.academia.edu/JunesangChoi">Junesang Choi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Far East Journal of Mathematical Sciences (FJMS), 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">Stirling numbers play an important role in various research subjects, in particular, in the combinatorial analysis. A number of identities involving Stirling numbers, harmonic numbers and generalized harmonic numbers have been established in various ways. Here we aim to present certain presumably new identities involving Stirling numbers of the first kind, harmonic and generalized harmonic numbers by analyzing the Beta integral function.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Certain Identities Involving Stirling Numbers of the First Kind","attachmentId":90265050,"attachmentType":"pdf","work_url":"https://www.academia.edu/85625135/Certain_Identities_Involving_Stirling_Numbers_of_the_First_Kind","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/85625135/Certain_Identities_Involving_Stirling_Numbers_of_the_First_Kind"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="56401022" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/56401022/On_Generalized_Stirling_Numbers_and_Polynomials">On Generalized Stirling Numbers and Polynomials</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="12292781" href="https://independent.academia.edu/DrHariSinghParihar">Dr. Hari Singh Parihar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</p><p class="ds-related-work--abstract ds2-5-body-sm">The object of this article is to present a generalization of stirling numbers and polynomials which were studied in a number of earlier work on the subject due to their importance for possible applications in certain problems arising in science and engineering (like curve fitting, coding theory, signal processing etc.). We prove that are result concerned the generalized stirling numbers</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Generalized Stirling Numbers and Polynomials","attachmentId":71806034,"attachmentType":"pdf","work_url":"https://www.academia.edu/56401022/On_Generalized_Stirling_Numbers_and_Polynomials","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/56401022/On_Generalized_Stirling_Numbers_and_Polynomials"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="126306035" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/126306035/Expression_and_Computation_of_Generalized_Stirling_Numbers">Expression and Computation of Generalized Stirling Numbers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44934279" href="https://independent.academia.edu/TianXiaoHe">Tian-Xiao He</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The journal of combinatorial mathematics and combinatorial computing, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Three algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Expression and Computation of Generalized Stirling Numbers","attachmentId":120203888,"attachmentType":"pdf","work_url":"https://www.academia.edu/126306035/Expression_and_Computation_of_Generalized_Stirling_Numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/126306035/Expression_and_Computation_of_Generalized_Stirling_Numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="84180874" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84180874/Generalized_Stirling_numbers_of_third_kind_and_its_applications_in_number_theory">Generalized Stirling numbers of third kind and its applications in number theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="140970888" href="https://independent.academia.edu/BrittoAntonyXavier">Britto Antony Xavier</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Journal of Mathematical Analysis, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, the authors generate the Stirling numbers of third kind to find formula for the sum of several types of product of polynomials and polynomial factorials using inverse of the generalized difference operator of n th kind in the field of numerical analysis. Suitable examples are provided to illustrate the main results.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Generalized Stirling numbers of third kind and its applications in number theory","attachmentId":89293737,"attachmentType":"pdf","work_url":"https://www.academia.edu/84180874/Generalized_Stirling_numbers_of_third_kind_and_its_applications_in_number_theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/84180874/Generalized_Stirling_numbers_of_third_kind_and_its_applications_in_number_theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="84159735" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84159735/Restricted_r_Stirling_numbers_and_their_combinatorial_applications">Restricted r-Stirling numbers and their combinatorial applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="20814662" href="https://gob.academia.edu/MiguelMendez">Miguel Mendez</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Applied Mathematics and Computation, 2019</p><p class="ds-related-work--abstract ds2-5-body-sm">We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Restricted r-Stirling numbers and their combinatorial applications","attachmentId":89279401,"attachmentType":"pdf","work_url":"https://www.academia.edu/84159735/Restricted_r_Stirling_numbers_and_their_combinatorial_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/84159735/Restricted_r_Stirling_numbers_and_their_combinatorial_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":89790571,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":89790571,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_89790571" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="103764767" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/103764767/Extended_Bernoulli_and_Stirling_matrices_and_related_combinatorial_identities">Extended Bernoulli and Stirling matrices and related combinatorial identities</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="64619535" href="https://independent.academia.edu/mustafada%C4%9Fl%C4%B12">mustafa dağlı</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Linear Algebra and its Applications, 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data-collection-position="4" data-entity-id="50930552" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50930552/Generalized_higher_order_Stirling_numbers">Generalized higher order Stirling numbers</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3853163" href="https://mansoura.academia.edu/BeihElDesouky">Beih El-Desouky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical and Computer Modelling, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Generalized higher order Stirling 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