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

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960

Offset: 0

Author

Gus Wiseman, Mar 13 2025

Keywords

• nonn
• more

Comments

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

Examples

			The a(1) = 1 through a(4) = 12 multisets:
  {1}  {1,2}    {1,2,3}        {1,2,3,4}
       {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
                {1,1,1,1,2}    {1,2,2,2,3}
                {1,1,1,1,1,1}  {1,1,1,1,2,4}
                               {1,1,1,2,2,3}
                               {1,1,1,1,1,1,4}
                               {1,1,1,1,1,2,3}
                               {1,1,1,1,2,2,2}
                               {1,1,1,1,1,1,1,3}
                               {1,1,1,1,1,1,2,2}
                               {1,1,1,1,1,1,1,1,2}
                               {1,1,1,1,1,1,1,1,1,1}
		

Crossrefs

The number of possible choices was A066843.

Multiset partitions into constant blocks: A006171, A279784, A295935.

Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.

Choosing divisors: A355747, A355733.

Sets of constant multisets with distinct sums: A381635, A381636, A381716.

Strict instead of constant partitions: A381808, A058694, A152827.

A000041 counts integer partitions, strict A000009, constant A000005.

A000688 counts multiset partitions into constant blocks.

A050361 and A381715 count multiset partitions into constant multisets.

A066723 counts partitions coarser than {1..n}, primorial case of A317141.

A265947 counts refinement-ordered pairs of integer partitions.

A321470 counts partitions finer than {1..n}, primorial case of A300383.

Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Programs

Mathematica

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]

Formula

Primorial case of A381453: a(n) = A381453(A002110(n)).

Extensions

a(16)-a(19) from Christian Sievers, Jun 04 2025