A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

1, 1, 2, 2, 5, 2, 6, 2, 10, 3, 6, 2, 24, 2, 6, 4, 17, 2, 36, 2, 18, 4, 6, 2, 86, 3, 6, 10, 18, 2, 44, 2, 50, 4, 6, 4, 159, 2, 6, 4, 62, 2, 44, 2, 18, 30, 6, 2, 486, 3, 12, 4, 18, 2, 140, 4, 62, 4, 6, 2, 932, 2, 6, 30, 157, 4, 44, 2, 18, 4, 20, 2, 1500, 2, 6

Offset: 0

Author

Gus Wiseman, Apr 03 2025

Keywords

• nonn

Comments

These are strict twice-partitions of weight n and type PRR.

Examples

			The a(1) = 1 through a(8) = 10 twice-partitions:
  (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
       (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
                    (1111)            (222)                (2222)
                    (11)(2)           (111111)             (22)(4)
                    (2)(11)           (111)(3)             (4)(22)
                                      (3)(111)             (1111)(4)
                                                           (4)(1111)
                                                           (11111111)
                                                           (1111)(22)
                                                           (22)(1111)
		

Links

• Gus Wiseman, Sequences enumerating triangles of integer partitions

Crossrefs

For distinct instead of equal block-sums we have A279786.

This is the strict case of A279789.

The orderless version is A304442, see A353833, A381995, A381871.

Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.

Partitions with no partition of this type are counted by A382076, strict case of A381993.

Normal multiset partitions of this type are counted by the strict case of A382204.

A006171 counts multiset partitions into constant blocks of integer partitions of n.

A050361 counts factorizations into distinct prime powers, see A381715.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.

Programs

Mathematica

Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]

Formula

a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!