%I #10 Feb 06 2024 19:31:57

%S 1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,2,0,0,0,1,0,0,1,3,5,0,0,0,3,5,0,0,1,5,

%T 17,11,0,0,0,4,20,21,0,0,1,9,53,80,34,0,0,0,6,60,167,91,0,0,1,11,121,

%U 418,410,87,0,0,0,10,149,816,1189,402

%N Triangle read by rows: T(n,k) is the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and no singletons or endpoints, 0 <= k <= floor(n/2).

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

%C T(n,k) is the number of binary matrices with n 1's, k rows and every row and column sum at least two up to permutation of rows and columns.

%H Andrew Howroyd, <a href="/A369927/b369927.txt">Table of n, a(n) for n = 0..675</a> (rows 0..50)

%F T(2*n,n) = A307316(n).

%e Triangle begins:

%e 1;

%e 0;

%e 0, 0;

%e 0, 0;

%e 0, 0, 1;

%e 0, 0, 0;

%e 0, 0, 1, 2;

%e 0, 0, 0, 1;

%e 0, 0, 1, 3, 5;

%e 0, 0, 0, 3, 5;

%e 0, 0, 1, 5, 17, 11;

%e 0, 0, 0, 4, 20, 21;

%e 0, 0, 1, 9, 53, 80, 34;

%e 0, 0, 0, 6, 60, 167, 91;

%e 0, 0, 1, 11, 121, 418, 410, 87;

%e 0, 0, 0, 10, 149, 816, 1189, 402;

%e ...

%e The T(4,2) = 1 partition is {{1,2},{1,2}}.

%e The corresponding matrix is:

%e [1 1]

%e [1 1]

%e The T(8,3) = 3 matrices are:

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

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

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

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o K(q, t, k)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g}

%o H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); K(q, t, k) - c*x}

%o G(n, y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}

%o T(n)={my(v=Vec(G(n, 'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}

%o { my(A=T(15)); for(i=1, #A, print(A[i])) }

%Y Row sums are A369926.

%Y Cf. A307316, A369286, A369287.

%K nonn,tabf

%O 0,16

%A _Andrew Howroyd_, Feb 06 2024