OFFSET
1,2
COMMENTS
Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..9
FORMULA
a(n) = Sum_{k=1..2^n-1} Stirling2(2^n-1,k) = Bell(2^n-1), 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^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_{k=1..2^n-1} Stirling2(2^n-1,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^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;
MATHEMATICA
BellB[2^Range[6]-1] (* Harvey P. Dale, Jul 22 2012 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Nov 18 2007
STATUS
approved