A combinatorial proof of symmetry among minimal star factorizations

TL;DR

This paper explains the symmetry in the number of minimal transitive star factorizations of permutations, showing it depends only on the conjugacy class, and provides a bijection between factorizations with different pivots.

Contribution

It offers a combinatorial proof and a bijection demonstrating the symmetry among minimal star factorizations based on conjugacy classes.

Findings

Number of minimal transitive star factorizations depends only on conjugacy class.

Established a bijection between factorizations with different pivots.

Clarified the role of the pivot in such factorizations.

Abstract

The number of minimal transitive star factorizations of a permutation was shown by Irving and Rattan to depend only on the conjugacy class of the permutation, a surprising result given that the pivot plays a very particular role in such factorizations. Here, we explain this symmetry and provide a bijection between minimal transitive star factorizations of a permutation \pi having pivot k and those having pivot k'.