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

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

Author

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

Keywords

• nonn

Comments

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

Links

• 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).

Crossrefs

Cf. A016269, A047707.

Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.

Cf. A094034, A094035, A094036, A094037.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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)