A356932 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

FORMULA

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

EXAMPLE

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}

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}]

PROG

(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

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.

Sequence in context: A296689 A327543 A096236 * A002574 A069765 A090427

Adjacent sequences: A356929 A356930 A356931 * A356933 A356934 A356935

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 11 2022

EXTENSIONS

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

STATUS

approved