A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612

Offset: 0

Author

Gus Wiseman, Mar 14 2025

Keywords

• nonn
• more

Examples

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}
		

Crossrefs

Set systems: A050342, A116539, A296120, A318361.

The number of possible choices was A152827, non-strict A058694.

Set multipartitions with distinct sums: A279785, A381718.

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

Choosing divisors: A355747, A355733.

Constant instead of strict partitions: A381807, A066843.

A000041 counts integer partitions, strict A000009, constant A000005.

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[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

Extensions

a(12)-a(16) from Christian Sievers, Jun 04 2025