1,2

Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).

a(n) = sum((stirling2(2^n-1,k)), k=1..2^n-1) = bell(2^n-1), where stirling2(n) is the Stirling number of the second kind and Bell(n) is the Bell number.

Let S={1,2,3,...,n} be a set of n elements and let

SU be the set of all nonempty subsets of S. The number of elements

of SU is |SU| = 2^n-1. Now form all possible set partitions from SU

where the empty set is excluded. This gives a set W and its number of

elements is |W| = sum((stirling2(2^n-1,k)), k=1..2^n-1).

For S={1,2} we have SU = { {1}, {2}, {1,2} } and W =

{

{{1}, {2}, {1, 2}},

{{1, 2}, {{1}, {2}}},

{{2}, {{1}, {1, 2}}},

{{1}, {{2}, {1, 2}}},

{{{1}, {2}, {1, 2}}}

}

and |W| = 5.

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n, nend, arg, k, w; nend:=5; for n from 1 to nend do arg:=2^n-1; w[n]:=sum((stirling2(arg, k)), k=1..arg); od; print(w[1], w[2], w[3], w[4], w[5], w[6], w[7], w[8], w[9], w[10]); end proc;

Cf. A000079, A000110, A008277, A077585, A135085.

Sequence in context: A051304 A060713 A085706 this_sequence A094630 A015940 A093853

Adjacent sequences: A135081 A135082 A135083 this_sequence A135085 A135086 A135087

nonn

Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 18 2007