The expected number of inversions after n adjacent transpositions

TL;DR

This paper derives a new formula for the expected number of inversions after applying n random adjacent transpositions in symmetric groups, analyzing its asymptotic behavior as parameters scale, based on a weighted walk model.

Contribution

It introduces a novel expression for the expected inversions and explores its asymptotics using a weighted walk framework, advancing understanding of permutation transpositions.

Findings

New explicit formula for expected inversions

Asymptotic analysis of inversions for large n and m

Connection to weighted walks in a triangular domain

Abstract

We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.