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 A319616 Number of non-isomorphic square multiset partitions of weight n. 87
 1, 1, 2, 4, 11, 27, 80, 230, 719, 2271, 7519, 25425, 88868, 317972, 1168360, 4392724, 16903393, 66463148, 266897917, 1093550522, 4568688612 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices. Also the number of square integer matrices with entries summing to n and no empty rows or columns, up to permutation of rows and columns. LINKS EXAMPLE Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions: 1: {{1}} 2: {{1,1}}    {{1},{2}} 3: {{1,1,1}}    {{1},{2,2}}    {{2},{1,2}}    {{1},{2},{3}} 4: {{1,1,1,1}}    {{1},{1,2,2}}    {{1},{2,2,2}}    {{2},{1,2,2}}    {{1,1},{2,2}}    {{1,2},{1,2}}    {{1,2},{2,2}}    {{1},{1},{2,3}}    {{1},{2},{3,3}}    {{1},{3},{2,3}}    {{1},{2},{3},{4}} Non-isomorphic representatives of the a(4) = 11 square matrices: . [4] . . [1 0]   [1 0]   [0 1]   [2 0]   [1 1]   [1 1] . [1 2]   [0 3]   [1 2]   [0 2]   [1 1]   [0 2] . . [1 0 0]   [1 0 0]   [1 0 0] . [1 0 0]   [0 1 0]   [0 0 1] . [0 1 1]   [0 0 2]   [0 1 1] . . [1 0 0 0] . [0 1 0 0] . [0 0 1 0] . [0 0 0 1] MATHEMATICA (* See A318795 for M[m, n, k]. *) T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n]; a[0] = 1; a[n_] := Sum[T[n, k], {k, 1, n}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* Jean-François Alcover, Nov 24 2018, after Andrew Howroyd *) PROG (PARI) \\ See A318795 for M. a(n) = {if(n==0, 1, sum(i=1, n, M(i, i, n) - 2*M(i, i-1, n) + M(i-1, i-1, n)))} \\ Andrew Howroyd, Nov 15 2018 CROSSREFS Row sums of A321615. Cf. A000219, A007716, A007718, A056156, A059201, A316980, A316983, A318795, A319560, A319616-A319646, A300913. Sequence in context: A123441 A086441 A316696 * A148130 A131482 A234845 Adjacent sequences:  A319613 A319614 A319615 * A319617 A319618 A319619 KEYWORD nonn,more AUTHOR Gus Wiseman, Sep 25 2018 EXTENSIONS a(11)-a(20) from Andrew Howroyd, Nov 15 2018 STATUS approved

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Last modified June 4 05:19 EDT 2020. Contains 334815 sequences. (Running on oeis4.)