login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006058 Number of connected labeled T_4-topologies with n points.
(Formerly M3030)
18
1, 1, 3, 16, 145, 2111, 47624, 1626003, 82564031, 6146805142, 662718022355, 102336213875523, 22408881211102698, 6895949927379360277, 2958271314760111914191, 1756322140048351303019576 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A349969 A109398 A294003 * A121588 A331538 A306397
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)