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A326878 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A326878 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A326878" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA326878">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A326878 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A326878 </div> <div class=seqname> Number of topologies whose points are a subset of {1..n}. </div> </div> <div class=scorerefs> 23 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291</div> <div class=seqdatalinks> (<a href="/A326878/list">list</a>; <a href="/A326878/graph">graph</a>; <a href="/search?q=A326878+-id:A326878">refs</a>; <a href="/A326878/listen">listen</a>; <a href="/history?seq=A326878">history</a>; <a href="/search?q=id:A326878&fmt=text">text</a>; <a href="/A326878/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A326878/b326878.txt">Table of n, a(n) for n=0..18.</a></div> <div class=sectline>Wikipedia <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Geoffrey_Critzer">Geoffrey Critzer</a>, Jul 12 2022: (Start)</div> <div class=sectline>E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>.</div> <div class=sectline>a(n) = Sum_{k=0..n} binomial(n,k)*<a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a>(k). (End)</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>The a(0) = 1 through a(2) = 7 topologies:</div> <div class=sectline> {{}} {{}} {{}}</div> <div class=sectline> {{},{1}} {{},{1}}</div> <div class=sectline> {{},{2}}</div> <div class=sectline> {{},{1,2}}</div> <div class=sectline> {{},{1},{1,2}}</div> <div class=sectline> {{},{2},{1,2}}</div> <div class=sectline> {{},{1},{2},{1,2}}</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&SubsetQ[#, Union[Union@@@Tuples[#, 2], Intersection@@@Tuples[#, 2]]]&]], {n, 0, 4}]</div> <div class=sectline>(* Second program: *)</div> <div class=sectline><a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a> = Cases[Import["https://oeis.org/<a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a>/b000798.txt", "Table"], {_, _}][[All, 2]];</div> <div class=sectline>a[n_] := Sum[Binomial[n, k]*<a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a>[[k+1]], {k, 0, n}];</div> <div class=sectline>a /@ Range[0, Length[<a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a>]-1] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Dec 30 2019 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Binomial transform of <a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a> (the covering case).</div> <div class=sectline>Cf. <a href="/A001035" title="Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).">A001035</a>, <a href="/A001930" title="Number of topologies, or transitive digraphs with n unlabeled nodes.">A001930</a>, <a href="/A003465" title="Number of ways to cover an n-set.">A003465</a>, <a href="/A014466" title="Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set.">A014466</a>, <a href="/A102896" title="Number of ACI algebras (or semilattices) on n generators with no annihilator.">A102896</a>, <a href="/A102897" title="Number of ACI algebras (or semilattices) on n generators.">A102897</a>, <a href="/A306445" title="Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.">A306445</a>, <a href="/A326866" title="Number of connectedness systems on n vertices.">A326866</a>, <a href="/A326876" title="BII-numbers of finite topologies without their empty set.">A326876</a>.</div> <div class=sectline>Sequence in context: <a href="/A066612" title="Number of primes between n^n and (n+1)^(n+1).">A066612</a> <a href="/A098637" title="Consider the family of multigraphs enriched by the species of odd sets. Sequence gives number of those multigraphs with n lo...">A098637</a> <a href="/A162045" title="Number of n X n binary arrays with rows and columns, considered as binary numbers, in nondecreasing order, and no more than ...">A162045</a> * <a href="/A153549" title="Number of planar n X n X n binary triangular grids with no more than 2 ones in any similarly oriented 2 X 2 X 2 subtriangle.">A153549</a> <a href="/A348880" title="G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(4*x)).">A348880</a> <a href="/A079491" title="Numerator of Sum_{k=0..n} binomial(n,k)/2^(k*(k-1)/2).">A079491</a></div> <div class=sectline>Adjacent sequences: <a href="/A326875" title="BII-numbers of set-systems that are closed under union.">A326875</a> <a href="/A326876" title="BII-numbers of finite topologies without their empty set.">A326876</a> <a href="/A326877" title="Number of connectedness systems covering n vertices without singletons.">A326877</a> * <a href="/A326879" title="BII-numbers of connected connectedness systems.">A326879</a> <a href="/A326880" title="BII-numbers of set-systems that are closed under nonempty intersection.">A326880</a> <a href="/A326881" title="Number of set-systems with {} that are closed under intersection and cover n vertices.">A326881</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Jul 30 2019</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 17 19:03 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>