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

Authors:
Volker Gebhardt
Affiliations:
School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith NSW 2751, Australia
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

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Abstract

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^-^1x2^n^-^1, we use matrices of size p(n)xp(n), where p(n) is the number of partitions of n.