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A369927 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).

%I A369927 #10 Feb 06 2024 19:31:57
%S A369927 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 A369927 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 A369927 418,410,87,0,0,0,10,149,816,1189,402
%N A369927 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 A369927 A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.
%C A369927 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 A369927 Andrew Howroyd, <a href="/A369927/b369927.txt">Table of n, a(n) for n = 0..675</a> (rows 0..50)
%F A369927 T(2*n,n) = A307316(n).
%e A369927 Triangle begins:
%e A369927   1;
%e A369927   0;
%e A369927   0, 0;
%e A369927   0, 0;
%e A369927   0, 0, 1;
%e A369927   0, 0, 0;
%e A369927   0, 0, 1,  2;
%e A369927   0, 0, 0,  1;
%e A369927   0, 0, 1,  3,   5;
%e A369927   0, 0, 0,  3,   5;
%e A369927   0, 0, 1,  5,  17,  11;
%e A369927   0, 0, 0,  4,  20,  21;
%e A369927   0, 0, 1,  9,  53,  80,   34;
%e A369927   0, 0, 0,  6,  60, 167,   91;
%e A369927   0, 0, 1, 11, 121, 418,  410,  87;
%e A369927   0, 0, 0, 10, 149, 816, 1189, 402;
%e A369927   ...
%e A369927 The T(4,2) = 1 partition is {{1,2},{1,2}}.
%e A369927 The corresponding matrix is:
%e A369927    [1 1]
%e A369927    [1 1]
%e A369927 The T(8,3) = 3 matrices are:
%e A369927    [1 1 1]  [1 1 1 0]  [1 1 1 1]
%e A369927    [1 1 1]  [1 1 0 1]  [1 1 0 0]
%e A369927    [1 1 0]  [0 0 1 1]  [0 0 1 1]
%o A369927 (PARI)
%o A369927 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o A369927 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 A369927 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 A369927 H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); K(q, t, k) - c*x}
%o A369927 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 A369927 T(n)={my(v=Vec(G(n, 'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
%o A369927 { my(A=T(15)); for(i=1, #A, print(A[i])) }
%Y A369927 Row sums are A369926.
%Y A369927 Cf. A307316, A369286, A369287.
%K A369927 nonn,tabf
%O A369927 0,16
%A A369927 _Andrew Howroyd_, Feb 06 2024