Computing growth functions of braid monoids and counting vertex-labelled bipartite graphs

We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type A(n-1) with respect to the simple elements (permutation braids) as generators. Instead of matrices of size 2(n-1) x 2(n-1), we use matrices of size p(n) x p(n), where p(n) is the number of partitions of n.