University of Minnesota Combinatorics Seminar
Friday, April 11, 2014
3:35pm in 570 Vincent Hall



Bijections and symmetries for factorizations of the long cycle

Alejandro Morales

LaCIM, University of Quebec at Montreal


Abstract

We study the factorizations of the permutation (1, 2, . . . , n) into k factors of given cycle types. Using the group algebra of the symmetric group, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases k = 2, 3, Schaeffer and Vassilieva gave a combinatorial proof of Jackson’s formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this talk to establish a series of bijections which unveil some of the combinatorial properties of these factorizations into k factors for all k. The first bijection is an instance of a correspondence of Bernardi between such factorizations and tree-rooted maps; certain graphs embedded on surfaces with a distinguished spanning tree. This is joint work with Olivier Bernardi.