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%I #10 Jan 11 2021 00:14:32
%S 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 14,63,147,154,110,47,22,1,14,102,278,409,329,194,73,30,1,20,150,568,
%U 991,1023,664,335,114,42,1,20,221,1020,2334,2844,2267,1243,549,170,56
%N 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 T(n,k) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts.
%H Andrew Howroyd, <a href="/A334550/b334550.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 2, 3;
%e 1, 5, 5, 5;
%e 1, 5, 12, 9, 7;
%e 1, 9, 23, 29, 17, 11;
%e 1, 9, 39, 62, 57, 28, 15;
%e 1, 14, 63, 147, 154, 110, 47, 22;
%e ...
%e The T(4,3) = 5 matrices are:
%e [1 0 0] [1 0 0] [1 1 0] [1 1 1] [1 1 0]
%e [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1]
%e [0 1 0] [0 1 1] [0 0 1]
%e [0 0 1]
%e The T(4,3) = 5 the set multipartitions are:
%e {{1,2}, {3}, {4}},
%e {{1,2}, {3}, {3}},
%e {{1,2}, {1}, {3}},
%e {{1,2}, {1}, {1}},
%e {{1,2}, {1}, {2}}.
%o (PARI) \\ See A321609 for definition of M.
%o T(n, k)={M(k, n, n) - M(k-1, n, n)}
%o for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print)
%o (PARI) \\ Faster version.
%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, n)={prod(j=1, #q, (1+x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
%o 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 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 { my(T=A(10)); for(n=1, #T, print(T[n,1..n])) }
%Y Row sums are A049311.
%Y Main diagonal is A000041.
%Y Cf. A317533, A321609.
%K nonn,tabl
%O 1,3
%A _Andrew Howroyd_, Jul 03 2020
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