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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

Table of n, a(n) for n=0..20.

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-Fran├žois 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 September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)