A137736 Number of set partitions of [n*(n-1)/2].

1, 1, 1, 5, 203, 115975, 1382958545, 474869816156751, 6160539404599934652455, 3819714729894818339975525681317, 139258505266263669602347053993654079693415, 359334085968622831041960188598043661065388726959079837

Offset: 0

Author

Thomas Wieder, Feb 09 2008

Keywords

• nonn

Comments

Among n persons we have (n^2-n)/2 undirected relations. We can set partition these relations into (up to) a(n) = Bell((n^2-n)/2) sets.

The number of graphs on n labeled nodes is A006125(n) = Sum_{k=0..(n^2-n)/2} binomial((n^2-n)/2,k).

See also A066655 which equals A066655(n) = Sum_{k=0..(n^2-n)/2} P((n^2-n)/2,k) where P(n) is the number of integer partitions of n.

See also A135084 = A000110(2^n-1) and A135085 = A000110(2^n).

Examples

			a(4) = Bell(6) = 203.
		

Crossrefs

Cf. A000110, A006125, A066655, A135084, A135085, A161680.

Programs

Maple

seq(combinat[bell](n*(n-1)/2), n=0..12);

Mathematica

a[n_]=BellB[n(n-1)/2];Array[a,12,0] (* James C. McMahon, Jun 02 2025 *)

Formula

a(n) = Bell(n*(n-1)/2) = A000110(n*(n-1)/2).

a(n) = Sum_{k=0..(n^2-n)/2} Stirling2((n^2-n)/2,k).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 24 2024