cp's OEIS Frontend

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

%I A382524 #6 Apr 03 2025 20:34:42
%S A382524 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,
%T A382524 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,
%U A382524 4,6,2,932,2,6,30,157,4,44,2,18,4,20,2,1500,2,6
%N A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.
%C A382524 These are strict twice-partitions of weight n and type PRR.
%H A382524 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F A382524 a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!
%e A382524 The a(1) = 1 through a(8) = 10 twice-partitions:
%e A382524   (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
%e A382524        (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
%e A382524                     (1111)            (222)                (2222)
%e A382524                     (11)(2)           (111111)             (22)(4)
%e A382524                     (2)(11)           (111)(3)             (4)(22)
%e A382524                                       (3)(111)             (1111)(4)
%e A382524                                                            (4)(1111)
%e A382524                                                            (11111111)
%e A382524                                                            (1111)(22)
%e A382524                                                            (22)(1111)
%t A382524 Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]
%Y A382524 For distinct instead of equal block-sums we have A279786.
%Y A382524 This is the strict case of A279789.
%Y A382524 The orderless version is A304442, see A353833, A381995, A381871.
%Y A382524 Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.
%Y A382524 Partitions with no partition of this type are counted by A382076, strict case of A381993.
%Y A382524 Normal multiset partitions of this type are counted by the strict case of A382204.
%Y A382524 A006171 counts multiset partitions into constant blocks of integer partitions of n.
%Y A382524 A050361 counts factorizations into distinct prime powers, see A381715.
%Y A382524 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A382524 Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.
%K A382524 nonn
%O A382524 0,3
%A A382524 _Gus Wiseman_, Apr 03 2025