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We derive a recurrence relation for the number of simple vertex-labelledbipartite graphs with given degrees of the vertices and use this result toobtain a new method for computing the growth function of the Artin monoid oftype $A_{n-1}$ with respect to the simple elements (permutation braids) asgenerators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we usematrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of$n$.

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