A006058 Number of connected labeled T_4-topologies with n points. (Formerly M3030)

1, 1, 3, 16, 145, 2111, 47624, 1626003, 82564031, 6146805142, 662718022355, 102336213875523, 22408881211102698, 6895949927379360277, 2958271314760111914191, 1756322140048351303019576

Offset

0,3

Comments

From Gus Wiseman, Aug 05 2019: (Start)

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:

{} {{1}} {{1,2}} {{1,2,3}}

{{1},{1,2}} {{1},{1,2,3}}

{{2},{1,2}} {{2},{1,2,3}}

{{3},{1,2,3}}

{{1,2},{1,2,3}}

{{1,3},{1,2,3}}

{{2,3},{1,2,3}}

{{1},{1,2},{1,2,3}}

{{1},{1,3},{1,2,3}}

{{2},{1,2},{1,2,3}}

{{2},{2,3},{1,2,3}}

{{3},{1,3},{1,2,3}}

{{3},{2,3},{1,2,3}}

{{1},{1,2},{1,3},{1,2,3}}

{{2},{1,2},{2,3},{1,2,3}}

{{3},{1,3},{2,3},{1,2,3}}

(End)

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Herman Jamke, Table of n, a(n) for n = 0..19

M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.

M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)

Formula

From Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008: (Start)

a(n) = Sum_{k=0..n-1} binomial(n, k)*A000798(k) if n>=1.

E.g.f.: Z4(x) = A(x)*(exp(x)-1) + 1 where A(x) denotes the e.g.f. for A000798. (End)

a(n) = A326909(n) - A000798(n). - Gus Wiseman, Aug 05 2019

Mathematica

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]]]];
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 *)
A000798 = Append[Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]], 0];
a[n_] := If[n == 0, 1, Sum[ Binomial[n, k] A000798[[k+1]], {k, 0, n-1}]];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Jan 01 2020 *)

Crossrefs

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

Cf. A001930, A003465, A108798, A306445, A326878, A326906.

Sequence in context: A349969 A109398 A294003 * A121588 A331538 A306397

Adjacent sequences: A006055 A006056 A006057 * A006059 A006060 A006061

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

Status

approved