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A356942
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Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.
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7
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1, 1, 4, 15, 61, 249, 1040, 4363, 18424, 78014, 331099, 1407080, 5985505, 25477399, 108493103, 462147381, 1969025286, 8390475609, 35757524184, 152398429323, 649555719160, 2768653475487, 11801369554033, 50304231997727, 214428538858889, 914039405714237
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OFFSET
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0,3
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COMMENTS
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A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(3) = 14 multiset partitions:
{{1}} {{1,1}} {{1,1,1}}
{{1,2}} {{1,1,2}}
{{1},{1}} {{1,2,2}}
{{1},{2}} {{1,2,3}}
{{1},{1,1}}
{{1},{1,2}}
{{1},{2,2}}
{{1},{2,3}}
{{2},{1,1}}
{{2},{1,2}}
{{3},{1,2}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{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]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Table[Length[Select[Join@@mps/@allnorm[n], And@@nogapQ/@#&]], {n, 0, 5}]
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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)}
R(n, k) = {EulerT(vector(n, j, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n, k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A, x, 1))} \\ Andrew Howroyd, Jan 01 2023
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CROSSREFS
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A011782 counts multisets covering an initial interval.
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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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