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%I #11 Dec 31 2022 11:19:59
%S 1,1,2,4,8,15,28,51,92,164,289,504,871,1493,2539,4290,7201,12017,
%T 19939,32911,54044,88330,143709,232817,375640,603755,966816,1542776,
%U 2453536,3889338,6146126,9683279,15211881,23830271,37230720,58015116,90174847,139820368,216286593
%N Number of multiset partitions of integer partitions of n with all distinct block sizes.
%H Andrew Howroyd, <a href="/A358836/b358836.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - _Andrew Howroyd_, Dec 31 2022
%e The a(1) = 1 through a(5) = 15 multiset partitions:
%e {1} {2} {3} {4} {5}
%e {1,1} {1,2} {1,3} {1,4}
%e {1,1,1} {2,2} {2,3}
%e {1},{1,1} {1,1,2} {1,1,3}
%e {1,1,1,1} {1,2,2}
%e {1},{1,2} {1,1,1,2}
%e {2},{1,1} {1},{1,3}
%e {1},{1,1,1} {1},{2,2}
%e {2},{1,2}
%e {3},{1,1}
%e {1,1,1,1,1}
%e {1},{1,1,2}
%e {2},{1,1,1}
%e {1},{1,1,1,1}
%e {1,1},{1,1,1}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
%o (PARI)
%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ _Andrew Howroyd_, Dec 31 2022
%Y The version for set partitions is A007837.
%Y For sums instead of sizes we have A271619.
%Y For constant instead of distinct sizes we have A319066.
%Y These multiset partitions are ranked by A326533.
%Y For odd instead of distinct sizes we have A356932.
%Y The version for twice-partitions is A358830.
%Y The case of distinct sums also is A358832.
%Y A001970 counts multiset partitions of integer partitions.
%Y A063834 counts twice-partitions, strict A296122.
%Y A336342 counts partitions of each part of a strict composition.
%Y Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 05 2022
%E Terms a(11) and beyond from _Andrew Howroyd_, Dec 31 2022
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