cp's OEIS Frontend

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

%I A381807 #11 Jun 04 2025 21:09:59
%S A381807 1,1,2,4,12,24,92,184,704,2016,7600,15200,80664,161328,601696,2198824,
%T A381807 9868544,19737088,102010480,204020960
%N A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.
%C A381807 A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.
%F A381807 Primorial case of A381453: a(n) = A381453(A002110(n)).
%e A381807 The a(1) = 1 through a(4) = 12 multisets:
%e A381807   {1}  {1,2}    {1,2,3}        {1,2,3,4}
%e A381807        {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
%e A381807                 {1,1,1,1,2}    {1,2,2,2,3}
%e A381807                 {1,1,1,1,1,1}  {1,1,1,1,2,4}
%e A381807                                {1,1,1,2,2,3}
%e A381807                                {1,1,1,1,1,1,4}
%e A381807                                {1,1,1,1,1,2,3}
%e A381807                                {1,1,1,1,2,2,2}
%e A381807                                {1,1,1,1,1,1,1,3}
%e A381807                                {1,1,1,1,1,1,2,2}
%e A381807                                {1,1,1,1,1,1,1,1,2}
%e A381807                                {1,1,1,1,1,1,1,1,1,1}
%t A381807 Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]
%Y A381807 The number of possible choices was A066843.
%Y A381807 Multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381807 Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
%Y A381807 Choosing divisors: A355747, A355733.
%Y A381807 Sets of constant multisets with distinct sums: A381635, A381636, A381716.
%Y A381807 Strict instead of constant partitions: A381808, A058694, A152827.
%Y A381807 A000041 counts integer partitions, strict A000009, constant A000005.
%Y A381807 A000688 counts multiset partitions into constant blocks.
%Y A381807 A050361 and A381715 count multiset partitions into constant multisets.
%Y A381807 A066723 counts partitions coarser than {1..n}, primorial case of A317141.
%Y A381807 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381807 A321470 counts partitions finer than {1..n}, primorial case of A300383.
%Y A381807 Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.
%K A381807 nonn,more
%O A381807 0,3
%A A381807 _Gus Wiseman_, Mar 13 2025
%E A381807 a(16)-a(19) from _Christian Sievers_, Jun 04 2025