A358837 Number of odd-length multiset partitions of integer partitions of n.

0, 1, 2, 4, 7, 14, 28, 54, 106, 208, 399, 757, 1424, 2642, 4860, 8851, 15991, 28673, 51095, 90454, 159306, 279067, 486598, 844514, 1459625, 2512227, 4307409, 7357347, 12522304, 21238683, 35903463, 60497684, 101625958, 170202949, 284238857, 473356564, 786196353

Offset: 0

Author

Gus Wiseman, Dec 05 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(5) = 14 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}}
                                 {{1},{1},{1,1}}  {{1,1,1,1,1}}
                                                  {{1},{1},{3}}
                                                  {{1},{2},{2}}
                                                  {{1},{1},{1,2}}
                                                  {{1},{2},{1,1}}
                                                  {{1},{1},{1,1,1}}
                                                  {{1},{1,1},{1,1}}
                                                  {{1},{1},{1},{1},{1}}
		

Links

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

Crossrefs

The version for set partitions is A024429.

These multiset partitions are ranked by A026424.

The version for partitions is A027193.

The version for twice-partitions is A358824.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions, strict A296122.

Cf. A000219, A141199, A336342, A358334, A358831.

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/@Reverse/@IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,10}]

PARI

P(v,y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v,1)) - Vec(P(v,-1)))/2} \\ Andrew Howroyd, Dec 31 2022

Formula

G.f.: ((1/Product_{k>=1} (1-x^k)^A000041(k)) - (1/Product_{k>=1} (1+x^k)^A000041(k))) / 2. - Andrew Howroyd, Dec 31 2022

Extensions

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