A combinatorial proof of symmetry among minimal star factorizations

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’.