A048143 Number of labeled connected simplicial complexes with n nodes.

1, 1, 1, 5, 84, 6348, 7743728, 2414572893530, 56130437190053299918162, 286386577668298410118121281812003477723571

Offset

0,4

Comments

Also number of connected antichains on a labeled n-set.

Links

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

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Greg Huber, Letters to N. J. A. Sloane, May 1983 [Annotated, corrected, scanned copy]

Harrison J. Piehowski, The Lean Number of a Hypergraph, arXiv:2607.14015 [math.CO], 2026. See p. 8 (Theorem 2.2).

Gus Wiseman, Every Clutter Is a Tree of Blobs, Mathematica J. 19, 2017.

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

Formula

Let u(n) = Sum_{j=0..n} ((-1)^j * binomial(n, j) * (A000372(n-j)-1)). Then a(0) = 1 and, for n >= 1, a(n) = u(n) - Sum_{k=1..n-1} (binomial(n-1, k-1) * a(k) * u(n-k)). - Robert P. P. McKone, Apr 15 2026

Example

For n=3 we could have 2 edges (in 3 ways), 3 edges (1 way), or 3 edges and a triangle (1 way), so a(3)=5.
a(5) = 1+75+645+1655+2005+1345+485+115+20+2 = 6348.

Mathematica

a000372Terms0To9={2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, 286386577668298411128469151667598498812366}; u[n_]:=u[n]=Sum[(-1)^j Binomial[n, j](a000372Terms0To9[[n-j+1]]-1), {j, 0, n}]; a[0]=1; a[n_]:=a[n]=u[n]-Sum[Binomial[n-1, k-1] a[k] u[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, Length[a000372Terms0To9]-1}] (* Robert P. P. McKone, Apr 14 2026 *)

Crossrefs

Cf. A000372.

Cf. A094033, A094034, A094035, A094036, A094037.

Cf. A000666, A006126, A030019, A054921, A120338, A275307, A285572.

Sequence in context: A288163 A268880 A367192 * A216420 A137083 A198586

Adjacent sequences: A048140 A048141 A048142 * A048144 A048145 A048146

Keyword

nonn,nice,more,changed

Author

Greg Huber, May 12 1983

Extensions

More terms from Vladeta Jovovic, Jun 17 2006

Entry revised by N. J. A. Sloane, Jul 27 2006

a(9) from Robert P. P. McKone, Apr 14 2026

Status

approved