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 A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size. 58
 1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n. Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018 From Gus Wiseman, Oct 11 2018: (Start) Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are: {{1,1,1,1}} {{1,1,2,2}} {{1,2,3,4}} {{1},{1,1,1}} {{1},{1,2,2}} {{1},{2,3,4}} {{1,1},{1,1}} {{1,1},{2,2}} {{1,2},{1,2}} {{1,2},{3,4}} {{1},{1},{1,1}} {{1},{1},{2,2}} {{1},{2},{1,2}} {{1},{2},{3,4}} {{1},{1},{1},{1}} {{1},{1},{2},{2}} {{1},{2},{3},{4}} (End) LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 FORMULA For p prime, a(p) = 2*A000041(p). a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020 EXAMPLE Non-isomorphic representatives of the a(4) = 17 multiset partitions: {{1,1,1,1}} {{1,1,2,2}} {{1,2,2,2}} {{1,2,3,3}} {{1,2,3,4}} {{1,1},{1,1}} {{1,1},{2,2}} {{1,2},{1,2}} {{1,2},{2,2}} {{1,2},{3,3}} {{1,2},{3,4}} {{1,3},{2,3}} {{1},{1},{1},{1}} {{1},{1},{2},{2}} {{1},{2},{2},{2}} {{1},{2},{3},{3}} {{1},{2},{3},{4}} MATHEMATICA permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m]; K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}]; RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!]; a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]]; Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *) PROG (PARI) \\ See A318951 for RowSumMats. a(n)={sumdiv(n, d, RowSumMats(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018 CROSSREFS Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951. Cf. A064573, A279787, A305551, A319616, A319056, A331485. Sequence in context: A365111 A190968 A127416 * A226631 A226634 A105271 Adjacent sequences: A306014 A306015 A306016 * A306018 A306019 A306020 KEYWORD nonn AUTHOR Gus Wiseman, Jun 17 2018 EXTENSIONS Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018 STATUS approved

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Last modified December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)