cp's OEIS Frontend

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

%I A356932 #10 Dec 30 2022 21:38:10
%S A356932 1,1,2,4,7,13,24,42,74,130,224,383,653,1100,1846,3079,5104,8418,13827,
%T A356932 22592,36774,59613,96271,154908,248441,397110,632823,1005445,1592962,
%U A356932 2516905,3966474,6235107,9777791,15297678,23880160,37196958,57819018,89691934,138862937
%N A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.
%H A356932 Andrew Howroyd, <a href="/A356932/b356932.txt">Table of n, a(n) for n = 0..1000</a>
%H A356932 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>
%F A356932 G.f.: 1/Product_{k>=1} (1 - x^k)^A027193(k). - _Andrew Howroyd_, Dec 30 2022
%e A356932 The a(1) = 1 through a(5) = 13 multiset partitions:
%e A356932   {1}  {2}     {3}        {4}           {5}
%e A356932        {1}{1}  {111}      {112}         {113}
%e A356932                {1}{2}     {1}{3}        {122}
%e A356932                {1}{1}{1}  {2}{2}        {1}{4}
%e A356932                           {1}{111}      {2}{3}
%e A356932                           {1}{1}{2}     {11111}
%e A356932                           {1}{1}{1}{1}  {1}{112}
%e A356932                                         {2}{111}
%e A356932                                         {1}{1}{3}
%e A356932                                         {1}{2}{2}
%e A356932                                         {1}{1}{111}
%e A356932                                         {1}{1}{1}{2}
%e A356932                                         {1}{1}{1}{1}{1}
%t A356932 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356932 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356932 Table[Length[Select[Join@@mps/@IntegerPartitions[n],OddQ[Times@@Length/@#]&]],{n,0,8}]
%o A356932 (PARI)
%o A356932 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A356932 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
%Y A356932 Partitions with odd multiplicities are counted by A055922.
%Y A356932 Odd-length multisets are counted by A000302, A027193, A058695, ranked by A026424.
%Y A356932 Other types: A050330, A356933, A356934, A356935.
%Y A356932 Other conditions: A001970, A006171, A007294, A089259, A107742, A356941.
%Y A356932 A000041 counts integer partitions, strict A000009.
%Y A356932 A001055 counts factorizations.
%Y A356932 Cf. A000670, A011782, A055887, A072233, A117958, A270995, A304969.
%K A356932 nonn
%O A356932 0,3
%A A356932 _Gus Wiseman_, Sep 11 2022
%E A356932 Terms a(13) and beyond from _Andrew Howroyd_, Dec 30 2022