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

%I #17 Jul 24 2024 09:47:16

%S 1,1,1,5,203,115975,1382958545,474869816156751,6160539404599934652455,

%T 3819714729894818339975525681317,

%U 139258505266263669602347053993654079693415,359334085968622831041960188598043661065388726959079837

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

%C Among n persons we have (n^2-n)/2 undirected relations. We can set partition these relations into (up to) A137736(n)=Bell((n^2-n)/2) sets.

%C The number of graphs on n labeled nodes is A006125(n)=sum(binomial((n^2-n)/2,k),k=0..(n^2-n)/2).

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

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

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

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

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

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

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

%K nonn

%O 0,4

%A _Thomas Wieder_, Feb 09 2008

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 24 2024