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%I #13 Aug 12 2022 09:23:25
%S 1,1,1,1,2,1,1,4,4,1,1,8,15,8,1,1,16,54,54,16,1,1,32,189,328,189,32,1,
%T 1,64,648,1856,1856,648,64,1,1,128,2187,9984,16145,9984,2187,128,1,1,
%U 256,7290,51712,129000,129000,51712,7290,256,1,1,512,24057,260096,968125,1475856,968125,260096,24057,512,1
%N Array read by antidiagonals: T(n,m) is the number of acyclic edge sets in the complete bipartite graph K_{n,m}.
%C In other words, the number of spanning forests of the complete bipartite graph K_{n,m} with isolated vertices allowed.
%H Andrew Howroyd, <a href="/A328887/b328887.txt">Table of n, a(n) for n = 0..1325</a>
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3235160/spanning-forests-of-bipartite-graphs-and-distinct-row-column-sums-of-binary-matr">Spanning forests of bipartite graphs and distinct row/column sums of binary matrices</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%F T(n,m) = 1 + Sum_{i=1..n} Sum_{j=1..m} binomial(n,i)*binomial(m,j)*A328888(i,j).
%e Array begins:
%e ====================================================================
%e n\m | 0 1 2 3 4 5 6 7
%e ----+---------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 2 4 8 16 32 64 128 ...
%e 2 | 1 4 15 54 189 648 2187 7290 ...
%e 3 | 1 8 54 328 1856 9984 51712 260096 ...
%e 4 | 1 16 189 1856 16145 129000 968125 6925000 ...
%e 5 | 1 32 648 9984 129000 1475856 15450912 151201728 ...
%e 6 | 1 64 2187 51712 968125 15450912 219682183 2862173104 ...
%e 7 | 1 128 7290 260096 6925000 151201728 2862173104 48658878080 ...
%e ...
%o (PARI) \\ here U is A328888 as matrix.
%o U(n, m=n)={my(M=matrix(n, m), N=matrix(n, m, n, m, n^(m-1) * m^(n-1))); for(n=1, n, for(m=1, m, M[n,m] = N[n,m] + sum(i=1, n-1, sum(j=1, m-1, binomial(n-1, i-1)*binomial(m, j)*N[i,j]*M[n-i, m-j])))); M}
%o T(n, m=n)={my(M=U(n, m)); matrix(n+1, m+1, n, m, 1 + sum(i=1, n-1, sum(j=1, m-1, binomial(n-1,i)*binomial(m-1,j)*M[i,j])))}
%o { my(A=T(7)); for(i=1, #A, print(A[i,])) }
%Y Column k=2 is A006234.
%Y Main diagonal is A297077.
%Y Cf. A072590, A328888.
%K nonn,tabl
%O 0,5
%A _Andrew Howroyd_, Oct 29 2019
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