Stochastic Couplings and Bijections from the Symmetric Group to Itself

TL;DR

This paper introduces four novel bijections from word sets to the symmetric group and explores their cycle structures, revealing complex behaviors of higher k-cycles through theoretical and empirical analysis.

Contribution

It presents new bijections inspired by stochastic processes and analyzes their cycle structures, including fixed points and higher k-cycles, combining theoretical and empirical methods.

Findings

Characterized properties of fixed points and k-cycles in the bijections

Demonstrated complexity of higher k-cycle structures empirically

Established connections between stochastic processes and symmetric group bijections

Abstract

Inspired by the Stochastic processes described by the Feller Coupling and Chinese Restaurant Processes, we create four different bijections from words in the set $[1]\times [2] \times\cdot \times[n]$ to $S_n$. We then compose these maps with their inverse to obtain a toal of six bijections $S_n \to S_n$. Following that, we investigate the fixed points ($1$-cycle) and higher $k$-cycles of these maps. We characterized some of their properties completely as well as empirically showing the complexity of the higher $k$-cycle structures for these maps.