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

TL;DR

This paper introduces a new method to compute the growth function of the Artin monoid of type A_{n-1} using partition-based matrices, simplifying calculations compared to previous matrix size approaches.

Contribution

It presents a recurrence relation for counting bipartite graphs and applies it to efficiently compute the growth function of specific braid monoids.

Findings

Derived a recurrence relation for bipartite graph enumeration

Developed a new matrix-based method for growth function computation

Reduced matrix size from exponential to partition number size

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 $2^{n-1}\times 2^{n-1}$, we use matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of $n$.