Degrees in random uniform minimal factorizations

TL;DR

This paper derives an explicit probability formula for the occurrence of specific elements in random uniform minimal factorizations of an n-cycle, using combinatorial bijections and generating functions.

Contribution

It introduces a novel explicit formula for joint probabilities in minimal factorizations, combining bijections with Cayley trees and generating function techniques.

Findings

Explicit joint probability formula for elements in minimal factorizations

Use of bijections with Cayley trees for combinatorial analysis

Explicit multivariate generating function computations

Abstract

We are interested in random uniform minimal factorizations of the $n$-cycle which are factorizations of $(1~2\dots n)$ into a product of $n-1$ transpositions. Our main result is an explicit formula for the joint probability that 1 and 2 appear a given number of times in a uniform minimal factorization. For this purpose, we combine bijections with Cayley trees together with explicit computations of multivariate generating functions.