A328887 Array read by antidiagonals: T(n,m) is the number of acyclic edge sets in the complete bipartite graph K_{n,m}.
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, 1, 64, 648, 1856, 1856, 648, 64, 1, 1, 128, 2187, 9984, 16145, 9984, 2187, 128, 1, 1, 256, 7290, 51712, 129000, 129000, 51712, 7290, 256, 1, 1, 512, 24057, 260096, 968125, 1475856, 968125, 260096, 24057, 512, 1
Offset: 0
Examples
Array begins: ==================================================================== n\m | 0 1 2 3 4 5 6 7 ----+--------------------------------------------------------------- 0 | 1 1 1 1 1 1 1 1 ... 1 | 1 2 4 8 16 32 64 128 ... 2 | 1 4 15 54 189 648 2187 7290 ... 3 | 1 8 54 328 1856 9984 51712 260096 ... 4 | 1 16 189 1856 16145 129000 968125 6925000 ... 5 | 1 32 648 9984 129000 1475856 15450912 151201728 ... 6 | 1 64 2187 51712 968125 15450912 219682183 2862173104 ... 7 | 1 128 7290 260096 6925000 151201728 2862173104 48658878080 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
- Mathematics Stack Exchange, Spanning forests of bipartite graphs and distinct row/column sums of binary matrices.
- Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Programs
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PARI
\\ here U is A328888 as matrix. 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} 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])))} { my(A=T(7)); for(i=1, #A, print(A[i,])) }
Formula
T(n,m) = 1 + Sum_{i=1..n} Sum_{j=1..m} binomial(n,i)*binomial(m,j)*A328888(i,j).
Comments