A307249 Number of simplicial complexes with n nodes.

1, 1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993

Offset

0,3

Comments

Except for a(0) = 1, this is also the number of antichains of nonempty sets covering n vertices (A006126). There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A261005. The non-covering case is A014466.

Links

Table of n, a(n) for n=0..9.

Francisco Ponce Carrión and Seth Sullivant, Marginal Independence and Partial Set Partitions, arXiv:2402.16292 [math.ST], 2024. See p. 21.

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Formula

Inverse binomial transform of A014466.

Example

Maximal simplices of the a(0) = 1 through a(3) = 9 simplicial complexes:
  {}    {{1}}  {{12}}    {{123}}
               {{1}{2}}  {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}

Mathematica

nn=5;
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[stableSets[Subsets[Range[n], {2, n}], SubsetQ]], {n, 0, nn}]

Crossrefs

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A317674, A319721, A320449.

Sequence in context: A337043 A008269 A039718 * A201381 A075538 A067965

Adjacent sequences: A307246 A307247 A307248 * A307250 A307251 A307252

Keyword

nonn,more

Author

Gus Wiseman, Mar 31 2019

Extensions

a(9) from Dmitry I. Ignatov, Nov 25 2023

Status

approved