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A322784
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Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
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6
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1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120
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OFFSET
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0,3
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COMMENTS
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A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020
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LINKS
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FORMULA
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EXAMPLE
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The a(1) = 1 through a(4) = 29 multiset partitions:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1,2}} {{1,2,3}} {{1,1,2,2}}
{{1},{1}} {{1},{1,1}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{1,1,1}}
{{2},{1,3}} {{1,1},{1,1}}
{{3},{1,2}} {{1},{1,2,2}}
{{1},{1},{1}} {{1,1},{2,2}}
{{1},{2},{3}} {{1,2},{1,2}}
{{1},{2,3,4}}
{{1,2},{3,4}}
{{1,3},{2,4}}
{{1,4},{2,3}}
{{2},{1,1,2}}
{{2},{1,3,4}}
{{3},{1,2,4}}
{{4},{1,2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,4}}
{{1},{4},{2,3}}
{{2},{2},{1,1}}
{{2},{3},{1,4}}
{{2},{4},{1,3}}
{{3},{4},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{3},{4}}
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MATHEMATICA
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u[n_, k_]:=u[n, k]=If[n==1, 1, Sum[u[n/d, d], {d, Select[Rest[Divisors[n]], #<=k&]}]];
Table[Sum[u[Array[Prime, d, 1, Times]^(n/d), Array[Prime, d, 1, Times]^(n/d)], {d, Divisors[n]}], {n, 12}]
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CROSSREFS
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Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.
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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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