A369926 - OEIS

%I #11 Feb 06 2024 19:32:01

%S 1,0,0,0,1,0,3,1,9,8,34,45,177,324,1048,2566,8050,22840,73562,231978,

%T 780221,2653042,9377141,33820014,125473936,475719042,1846424607,

%U 7317819857,29611827086,122190972442,513900819816,2201109101784,9595815668795,42553843201446,191861748624324,879049648551947

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

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

%C 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.

%H Andrew Howroyd, <a href="/A369926/b369926.txt">Table of n, a(n) for n = 0..50</a>

%e The a(8) = 9 matrices are:

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

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

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

%e .

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

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

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

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

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

%Y Row sums of A369927.

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

%Y Cf. A330055 (set-systems).

%K nonn

%O 0,7

%A _Andrew Howroyd_, Feb 06 2024