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

0,3

Comments

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

Links

Table of n, a(n) for n=0..74.

Gus Wiseman, Sequences enumerating triangles of integer partitions

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A011143 A240081 A305791 * A299764 A301830 A305799

Adjacent sequences: A382521 A382522 A382523 * A382525 A382526 A382527

Keyword

nonn

Author

Gus Wiseman, Apr 03 2025

Status

approved