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A334550 Triangle read by rows: T(n,k) is the number of binary matrices with n ones, k columns and no zero rows or columns, up to permutations of rows and columns.

%I A334550 #10 Jan 11 2021 00:14:32
%S A334550 1,1,2,1,2,3,1,5,5,5,1,5,12,9,7,1,9,23,29,17,11,1,9,39,62,57,28,15,1,
%T A334550 14,63,147,154,110,47,22,1,14,102,278,409,329,194,73,30,1,20,150,568,
%U A334550 991,1023,664,335,114,42,1,20,221,1020,2334,2844,2267,1243,549,170,56
%N A334550 Triangle read by rows: T(n,k) is the number of binary matrices with n ones, k columns and no zero rows or columns, up to permutations of rows and columns.
%C A334550 T(n,k) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts.
%H A334550 Andrew Howroyd, <a href="/A334550/b334550.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e A334550 Triangle begins:
%e A334550   1;
%e A334550   1,  2;
%e A334550   1,  2,  3;
%e A334550   1,  5,  5,   5;
%e A334550   1,  5, 12,   9,   7;
%e A334550   1,  9, 23,  29,  17,  11;
%e A334550   1,  9, 39,  62,  57,  28, 15;
%e A334550   1, 14, 63, 147, 154, 110, 47, 22;
%e A334550   ...
%e A334550 The T(4,3) = 5 matrices are:
%e A334550   [1 0 0]   [1 0 0]   [1 1 0]   [1 1 1]   [1 1 0]
%e A334550   [1 0 0]   [1 0 0]   [1 0 0]   [1 0 0]   [1 0 1]
%e A334550   [0 1 0]   [0 1 1]   [0 0 1]
%e A334550   [0 0 1]
%e A334550 The T(4,3) = 5 the set multipartitions are:
%e A334550   {{1,2}, {3}, {4}},
%e A334550   {{1,2}, {3}, {3}},
%e A334550   {{1,2}, {1}, {3}},
%e A334550   {{1,2}, {1}, {1}},
%e A334550   {{1,2}, {1}, {2}}.
%o A334550 (PARI) \\ See A321609 for definition of M.
%o A334550 T(n, k)={M(k, n, n) - M(k-1, n, n)}
%o A334550 for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print)
%o A334550 (PARI) \\ Faster version.
%o A334550 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 A334550 K(q, t, n)={prod(j=1, #q, (1+x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
%o A334550 G(m,n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
%o A334550 A(n,m=n)={my(p=sum(k=0, m, G(k,n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
%o A334550 { my(T=A(10)); for(n=1, #T, print(T[n,1..n])) }
%Y A334550 Row sums are A049311.
%Y A334550 Main diagonal is A000041.
%Y A334550 Cf. A317533, A321609.
%K A334550 nonn,tabl
%O A334550 1,3
%A A334550 _Andrew Howroyd_, Jul 03 2020