A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Example

The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{1,1}
                                      {1,1,1,1,1}
                                      {1},{1,1,2}
                                      {2},{1,1,1}
                                      {1},{1,1,1,1}
                                      {1,1},{1,1,1}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n], UnsameQ@@Length/@#&]], {n, 0, 10}]

Prog

(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n, y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

Crossrefs

The version for set partitions is A007837.

For sums instead of sizes we have A271619.

For constant instead of distinct sizes we have A319066.

These multiset partitions are ranked by A326533.

For odd instead of distinct sizes we have A356932.

The version for twice-partitions is A358830.

The case of distinct sums also is A358832.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions, strict A296122.

A336342 counts partitions of each part of a strict composition.

Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.

Sequence in context: A007673 A182725 A334635 * A029907 A005682 A114833

Adjacent sequences: A358833 A358834 A358835 * A358837 A358838 A358839

Keyword

nonn

Author

Gus Wiseman, Dec 05 2022

Extensions

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

Status

approved