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A369286 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no constant parts or vertices that appear in only one part, 0 <= k <= floor(n/2).

%I A369286 #8 Jan 28 2024 18:10:45
%S A369286 1,0,0,0,0,0,0,0,1,0,0,1,0,0,5,2,0,0,6,3,0,0,16,16,5,0,0,22,44,13,0,0,
%T A369286 45,135,82,11,0,0,64,338,301,52,0,0,119,880,1233,382,34,0,0,171,2024,
%U A369286 4090,1936,211,0,0,294,4674,13474,9500,1843,87,0,0,433,10191,40532,40817,11778,873
%N A369286 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no constant parts or vertices that appear in only one part, 0 <= k <= floor(n/2).
%C A369286 T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column having at least two nonzero entries up to permutation of rows and columns.
%H A369286 Andrew Howroyd, <a href="/A369286/b369286.txt">Table of n, a(n) for n = 0..675</a> (rows 0..50)
%F A369286 T(2*n,n) = A307316(n).
%e A369286 Triangle begins:
%e A369286   1;
%e A369286   0;
%e A369286   0, 0;
%e A369286   0, 0;
%e A369286   0, 0,   1;
%e A369286   0, 0,   1;
%e A369286   0, 0,   5,    2;
%e A369286   0, 0,   6,    3;
%e A369286   0, 0,  16,   16,    5;
%e A369286   0, 0,  22,   44,   13;
%e A369286   0, 0,  45,  135,   82,   11;
%e A369286   0, 0,  64,  338,  301,   52;
%e A369286   0, 0, 119,  880, 1233,  382,  34;
%e A369286   0, 0, 171, 2024, 4090, 1936, 211;
%e A369286   ...
%e A369286 The T(6,2) = 5 multiset partitions are:
%e A369286   {{1,1,1,2}, {1,2}},
%e A369286   {{1,1,2,2}, {1,2}},
%e A369286   {{1,1,2}, {1,1,2}},
%e A369286   {{1,1,2}, {1,2,2}},
%e A369286   {{1,2,3}, {1,2,3}}.
%e A369286 The corresponding T(6,2) = 5 matrices are:
%e A369286   [3 1]  [2 2]  [2 1]  [2 1]  [1 1 1]
%e A369286   [1 1]  [1 1]  [2 1]  [1 2]  [1 1 1]
%e A369286 The T(6,3) = 2 matrices are:
%e A369286   [1 1]  [1 1 0]
%e A369286   [1 1]  [1 0 1]
%e A369286   [1 1]  [0 1 1]
%o A369286 (PARI)
%o A369286 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A369286 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 A369286 K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
%o A369286 H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); eta(x + O(x*x^k))*(1 + x*Ser(K(q,t,k))) + x*(1-c)/(1-x) - 1}
%o A369286 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 A369286 T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
%o A369286 { my(A=T(15)); for(i=1, #A, print(A[i])) }
%Y A369286 Row sums are A321760.
%Y A369286 Cf. A307316, A369287.
%K A369286 nonn,tabf
%O A369286 0,15
%A A369286 _Andrew Howroyd_, Jan 28 2024