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A326517
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Number of normal multiset partitions of weight n where each part has a different size.
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12
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1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856
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OFFSET
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0,3
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COMMENTS
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A multiset partition is normal if it covers an initial interval of positive integers.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(3) = 12 normal multiset partitions:
{} {{1}} {{1,1}} {{1,1,1}}
{{1,2}} {{1,1,2}}
{{1,2,2}}
{{1,2,3}}
{{1},{1,1}}
{{1},{1,2}}
{{1},{2,2}}
{{1},{2,3}}
{{2},{1,1}}
{{2},{1,2}}
{{2},{1,3}}
{{3},{1,2}}
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
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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]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n], UnsameQ@@Length/@#&]], {n, 0, 6}]
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PROG
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(PARI)
R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020
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CROSSREFS
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Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.
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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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