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

0,3

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.

Links

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

Example

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}.

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 *)

Crossrefs

Cf. A102356, A102456.

Sequence in context: A062378 A073753 A290602 * A078090 A174341 A168516

Adjacent sequences: A255401 A255402 A255403 * A255405 A255406 A255407

Keyword

nonn

Author

Andrei Cretu, Feb 22 2015

Extensions

More terms from Alois P. Heinz, Feb 25 2015

Status

approved