A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.
1, 1, 3, 5, 10, 14, 25, 33, 54, 73, 107, 140, 207, 264, 369, 479, 652, 828, 1112, 1400, 1848, 2326, 3009, 3762, 4856, 6020, 7648, 9478, 11942, 14705, 18427, 22576, 28083, 34350, 42429, 51714, 63680, 77289, 94618, 114648, 139773, 168799, 205144, 247128, 299310, 359958, 434443, 521255, 627812, 751665, 902862
Offset: 0
Keywords
Examples
The a(1) = 1 through a(6) = 25 multiset partitions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(1)(1) (111) (22) (23) (24)
(1)(11) (112) (113) (33)
(1)(1)(1) (1111) (122) (114)
(2)(2) (1112) (123)
(1)(111) (11111) (222)
(11)(11) (2)(12) (1113)
(1)(1)(11) (1)(1111) (1122)
(1)(1)(1)(1) (11)(111) (3)(3)
(1)(1)(111) (11112)
(1)(11)(11) (111111)
(1)(1)(1)(11) (12)(12)
(1)(1)(1)(1)(1) (2)(112)
(2)(2)(2)
(1)(11111)
(11)(1111)
(111)(111)
(1)(1)(1111)
(1)(11)(111)
(11)(11)(11)
(1)(1)(1)(111)
(1)(1)(11)(11)
(1)(1)(1)(1)(11)
(1)(1)(1)(1)(1)(1)
Crossrefs
Programs
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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],SameQ@@Times@@@#&]],{n,8}] -
PARI
G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M} a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ Andrew Howroyd, Oct 26 2018
Extensions
a(13)-a(50) from Andrew Howroyd, Oct 26 2018