A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

1, 1, 2, 4, 7, 13, 24, 42, 74, 130, 224, 383, 653, 1100, 1846, 3079, 5104, 8418, 13827, 22592, 36774, 59613, 96271, 154908, 248441, 397110, 632823, 1005445, 1592962, 2516905, 3966474, 6235107, 9777791, 15297678, 23880160, 37196958, 57819018, 89691934, 138862937

Offset: 0

Author

Gus Wiseman, Sep 11 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(5) = 13 multiset partitions:
  {1}  {2}     {3}        {4}           {5}
       {1}{1}  {111}      {112}         {113}
               {1}{2}     {1}{3}        {122}
               {1}{1}{1}  {2}{2}        {1}{4}
                          {1}{111}      {2}{3}
                          {1}{1}{2}     {11111}
                          {1}{1}{1}{1}  {1}{112}
                                        {2}{111}
                                        {1}{1}{3}
                                        {1}{2}{2}
                                        {1}{1}{111}
                                        {1}{1}{1}{2}
                                        {1}{1}{1}{1}{1}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Crossrefs

Partitions with odd multiplicities are counted by A055922.

Odd-length multisets are counted by A000302, A027193, A058695, ranked by A026424.

Other types: A050330, A356933, A356934, A356935.

Other conditions: A001970, A006171, A007294, A089259, A107742, A356941.

A000041 counts integer partitions, strict A000009.

A001055 counts factorizations.

Cf. A000670, A011782, A055887, A072233, A117958, A270995, A304969.

Programs

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],OddQ[Times@@Length/@#]&]],{n,0,8}]

PARI

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(1/prod(k=1, n, (1 - x^k + O(x*x^n))^u[k])) } \\ Andrew Howroyd, Dec 30 2022

Formula

G.f.: 1/Product_{k>=1} (1 - x^k)^A027193(k). - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(13) and beyond from Andrew Howroyd, Dec 30 2022