A255404 Number of different integer partitions of n that produce the maximum number of set partitions for a set of cardinality n.

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 3, 2, 1, 4, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 6, 4, 1, 2, 1, 5, 5, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 5, 2, 2, 1, 1, 4, 1, 1, 2, 3, 1, 8, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 2, 1, 1, 1, 1

Offset: 0

Author

Andrei Cretu, Feb 22 2015

Keywords

• nonn

Comments

If n=Sum_i[n_i], the number of set partitions can be written as sp=n!/Prod_i,j(n_i!m_j!) where m_j is the multiplicity of the integer j in the n_i's. For certain integers, this number is maximized by more than one partition.

Examples

			For n=9, {1,1,2,2,3} maximizes the number of set partitions, while for n=10, this number is maximized by {1,2,3,4}, {1,1,2,3,3}, {1,2,2,2,3} and {1,1,1,2,2,3}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A102356, A102456.

Programs

Mathematica

Prod[l_] := Apply[Times, Map[#! &, l]]*
    Apply[Times, Map[Count[l, #]! &, Range[Max[Length[l]]]]]
b[n_] := (Min[Map[Prod, IntegerPartitions[n]]])
a[n_] := Count[Map[Prod, IntegerPartitions[n]], b[n]]
Table[a[n], {n, 0, 20}] (* after A102356 *)

Extensions

More terms from Alois P. Heinz, Feb 25 2015