




OFFSET

1,2


COMMENTS

Number of set partitions of all nonempty subsets of a set, Bell(2^n1).


LINKS



FORMULA

a(n) = Sum_{k=1..2^n1} Stirling2(2^n1,k) = Bell(2^n1), where Stirling2(n, k) is the Stirling number of the second kind and Bell(n) is the Bell number.


EXAMPLE

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^n1. 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_{k=1..2^n1} Stirling2(2^n1,k).
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.


MAPLE

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n, nend, arg, k, w; nend:=5; for n from 1 to nend do arg:=2^n1; 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;


MATHEMATICA



PROG

(Python)
from sympy import bell


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



