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
Andrew Howroyd, Table of n, a(n) for n = 0..50
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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 25 2018
EXTENSIONS
a(11)-a(20) from Andrew Howroyd, Nov 15 2018
a(21) onwards from Andrew Howroyd, Jan 15 2024
STATUS
approved