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A317775
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Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
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7
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1, 3, 10, 36, 136, 596, 2656, 13187, 68226, 381572, 2233091, 13940407, 90981030, 626911429, 4509031955, 33987610040, 266668955183, 2180991690286, 18512572760155, 163103174973092, 1487228204311039, 14027782824491946, 136585814043190619, 1371822048393658001, 14190528438090988629
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OFFSET
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1,2
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LINKS
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EXAMPLE
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The a(3) = 10 strict multiset partitions:
{{1,1,1}}, {{1},{1,1}},
{{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
{{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.
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MATHEMATICA
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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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n], UnsameQ@@#&]], {n, 6}]
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PROG
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(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p, n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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