A319616 Number of non-isomorphic square multiset partitions of weight n.

1, 1, 2, 4, 11, 27, 80, 230, 719, 2271, 7519, 25425, 88868, 317972, 1168360, 4392724, 16903393, 66463148, 266897917, 1093550522, 4568688612, 19448642187, 84308851083, 371950915996, 1669146381915, 7615141902820, 35304535554923, 166248356878549, 794832704948402, 3856672543264073, 18984761300310500

Offset: 0

Author

Gus Wiseman, Sep 25 2018

Keywords

• nonn

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Row sums of A321615.

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

Programs

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 *)

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

PARI

\\ See A340652 for G.
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

Extensions

a(11)-a(20) from Andrew Howroyd, Nov 15 2018

a(21) onwards from Andrew Howroyd, Jan 15 2024