A094034 -id:A094034 - cp's OEIS frontend

A048143 Number of labeled connected simplicial complexes with n nodes.

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

Offset: 0

Greg Huber, May 12 1983

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

			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.

Patrick De Causmaecker, 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]
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

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

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

More terms from Vladeta Jovovic, Jun 17 2006

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

A094033 Number of connected 2-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 3, 18, 75, 270, 903, 2898, 9075, 27990, 85503, 259578, 784875, 2366910, 7125303, 21425058, 64373475, 193317030, 580344303, 1741819338, 5227030875, 15684238350, 47059006503, 141189602418, 423593973075, 1270832250870

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 10 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 10.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Adam Roman, Igor T. Podolak and Agnieszka Deszynska, On the number of clusterings in a hierarchical classification model with overlapping clusters, Schedae Informaticae, Volume 20, 2011.
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

Cf. A016269, A047707.
Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.
Cf. A094034, A094035, A094036, A094037.

[seq(stirling2(n,3)*3,n=0..26)]; # Zerinvary Lajos, Dec 06 2006

Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)

x='x+O('x^50); concat([0,0,0],Vec(serlaplace((exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!))) \\ G. C. Greubel, Oct 06 2017

a(n) = 3 * A000392(n).
E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
From Colin Barker, Mar 31 2012: (Start)
a(n) = (3^n-3*2^n+3)/2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)

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A048143 Number of labeled connected simplicial complexes with n nodes.

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A094033 Number of connected 2-element antichains on a labeled n-set.

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