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A006058 Number of connected labeled T_4-topologies with n points.

%I A006058 M3030 #31 Jan 01 2020 15:21:30
%S A006058 1,1,3,16,145,2111,47624,1626003,82564031,6146805142,662718022355,
%T A006058 102336213875523,22408881211102698,6895949927379360277,
%U A006058 2958271314760111914191,1756322140048351303019576
%N A006058 Number of connected labeled T_4-topologies with n points.
%C A006058 From _Gus Wiseman_, Aug 05 2019: (Start)
%C A006058 For n > 0, also the number of topologies covering {1..n} whose nonempty open sets have nonempty intersection. Also the number of topologies covering {1..n} whose nonempty open sets are pairwise intersecting. For example, the a(0) = 1 through a(3) = 16 topologies (empty sets not shown) are:
%C A006058   {}  {{1}}  {{1,2}}      {{1,2,3}}
%C A006058              {{1},{1,2}}  {{1},{1,2,3}}
%C A006058              {{2},{1,2}}  {{2},{1,2,3}}
%C A006058                           {{3},{1,2,3}}
%C A006058                           {{1,2},{1,2,3}}
%C A006058                           {{1,3},{1,2,3}}
%C A006058                           {{2,3},{1,2,3}}
%C A006058                           {{1},{1,2},{1,2,3}}
%C A006058                           {{1},{1,3},{1,2,3}}
%C A006058                           {{2},{1,2},{1,2,3}}
%C A006058                           {{2},{2,3},{1,2,3}}
%C A006058                           {{3},{1,3},{1,2,3}}
%C A006058                           {{3},{2,3},{1,2,3}}
%C A006058                           {{1},{1,2},{1,3},{1,2,3}}
%C A006058                           {{2},{1,2},{2,3},{1,2,3}}
%C A006058                           {{3},{1,3},{2,3},{1,2,3}}
%C A006058 (End)
%D A006058 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006058 Herman Jamke, <a href="/A006058/b006058.txt">Table of n, a(n) for n = 0..19</a>
%H A006058 M. Erné, <a href="http://dx.doi.org/10.1007/BF01173716">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259.
%H A006058 M. Erné, <a href="/A006056/a006056.pdf">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
%F A006058 From Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008: (Start)
%F A006058 a(n) = Sum_{k=0..n-1} binomial(n, k)*A000798(k) if n>=1.
%F A006058 E.g.f.: Z4(x) = A(x)*(exp(x)-1) + 1 where A(x) denotes the e.g.f. for A000798. (End)
%F A006058 a(n) = A326909(n) - A000798(n). - _Gus Wiseman_, Aug 05 2019
%t A006058 stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t A006058 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}] (* _Gus Wiseman_, Aug 05 2019 *)
%t A006058 A000798 = Append[Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]], 0];
%t A006058 a[n_] := If[n == 0, 1, Sum[ Binomial[n, k] A000798[[k+1]], {k, 0, n-1}]];
%t A006058 a /@ Range[0, Length[A000798]-1] (* _Jean-François Alcover_, Jan 01 2020 *)
%Y A006058 Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
%Y A006058 Cf. A001930, A003465, A108798, A306445, A326878, A326906.
%K A006058 nonn,nice
%O A006058 0,3
%A A006058 _N. J. A. Sloane_
%E A006058 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008