A combinatorial proof of symmetry among minimal star factorizations

Bridget Eileen Tenner

Published 2011-09-21, updated 2012-05-19Version 2

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'.