A319616 - OEIS

%I #24 Jan 16 2024 22:05:05

%S 1,1,2,4,11,27,80,230,719,2271,7519,25425,88868,317972,1168360,

%T 4392724,16903393,66463148,266897917,1093550522,4568688612,

%U 19448642187,84308851083,371950915996,1669146381915,7615141902820,35304535554923,166248356878549,794832704948402,3856672543264073,18984761300310500

%N Number of non-isomorphic square multiset partitions of weight n.

%C A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices.

%C 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.

%H Andrew Howroyd, <a href="/A319616/b319616.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:

%e 1: {{1}}

%e 2: {{1,1}}

%e    {{1}, {2}}

%e 3: {{1,1,1}}

%e    {{1}, {2,2}}

%e    {{2}, {1,2}}

%e    {{1}, {2},{3}}

%e 4: {{1,1,1,1}}

%e    {{1}, {1,2,2}}

%e    {{1}, {2,2,2}}

%e    {{2}, {1,2,2}}

%e    {{1,1}, {2,2}}

%e    {{1,2}, {1,2}}

%e    {{1,2}, {2,2}}

%e    {{1}, {1}, {2,3}}

%e    {{1}, {2}, {3,3}}

%e    {{1}, {3}, {2,3}}

%e    {{1}, {2}, {3}, {4}}

%e Non-isomorphic representatives of the a(4) = 11 square matrices:

%e . [4]

%e .

%e . [1 0]   [1 0]   [0 1]   [2 0]   [1 1]   [1 1]

%e . [1 2]   [0 3]   [1 2]   [0 2]   [1 1]   [0 2]

%e .

%e . [1 0 0]   [1 0 0]   [1 0 0]

%e . [1 0 0]   [0 1 0]   [0 0 1]

%e . [0 1 1]   [0 0 2]   [0 1 1]

%e .

%e . [1 0 0 0]

%e . [0 1 0 0]

%e . [0 0 1 0]

%e . [0 0 0 1]

%t (* See A318795 for M[m, n, k]. *)

%t T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];

%t a[0] = 1; a[n_] := Sum[T[n, k], {k, 1, n}];

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* _Jean-François Alcover_, Nov 24 2018, after _Andrew Howroyd_ *)

%o (PARI) \\ See A318795 for M.

%o 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

%o (PARI) \\ See A340652 for G.

%o seq(n)={Vec(1 + sum(k=1,n,polcoef(G(k,n,n,y),k,y) - polcoef(G(k-1,n,n,y),k,y)))} \\ _Andrew Howroyd_, Jan 15 2024

%Y Row sums of A321615.

%Y Cf. A000219, A007716, A007718, A056156, A059201, A316980, A316983, A318795, A319560, A319616-A319646, A300913.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 25 2018

%E a(11)-a(20) from _Andrew Howroyd_, Nov 15 2018

%E a(21) onwards from _Andrew Howroyd_, Jan 15 2024