A369926 Number of non-isomorphic set multipartitions (multisets of sets) of weight n without endpoints or singletons.

1, 0, 0, 0, 1, 0, 3, 1, 9, 8, 34, 45, 177, 324, 1048, 2566, 8050, 22840, 73562, 231978, 780221, 2653042, 9377141, 33820014, 125473936, 475719042, 1846424607, 7317819857, 29611827086, 122190972442, 513900819816, 2201109101784, 9595815668795, 42553843201446, 191861748624324, 879049648551947

Offset: 0

Author

Andrew Howroyd, Feb 06 2024

Keywords

• nonn

Comments

A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.

a(n) is also the number of binary matrices with a total of n 1's and every row and column sum at least 2 up to permutation of rows and columns.

Examples

			The a(8) = 9 matrices are:
   [1 1 1 1]  [1 1 1]  [1 1 1 0]  [1 1 1 1]
   [1 1 1 1]  [1 1 1]  [1 1 0 1]  [1 1 0 0]
              [1 1 0]  [0 0 1 1]  [0 0 1 1]
.
   [1 1]  [1 1 0]  [1 1 0]  [1 1 0 0]  [1 1 0 0]
   [1 1]  [1 1 0]  [1 1 0]  [1 1 0 0]  [1 0 1 0]
   [1 1]  [1 0 1]  [1 0 1]  [0 0 1 1]  [0 1 0 1]
   [1 1]  [1 0 1]  [0 1 1]  [0 0 1 1]  [0 0 1 1]
		

Links

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

Crossrefs

Row sums of A369927.

A321677 is the case without singletons but allowing endpoints (or by duality without endpoints but allowing singletons).

Cf. A330055 (set-systems).

Programs

PARI

Vec(G(25,1)) \\ G defined in A369927.