Bijections and symmetries for the factorizations of the long cycle

Olivier Bernardi (1) , Alejandro H. Morales (1)

1 MIT - Massachusetts Institute of Technology

We study the factorizations of the permutation $(1,2,\ldots,n)$ into $k$ factors of given cycle types. Using representation theory, 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 article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of $(1,2,\ldots,n)$ into $k$ factors for all $k$. We thereby obtain refinements of Jackson's formulas which extend the cases $k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of ''constellations'', which are graphs embedded in surfaces encoding the transitive factorizations of permutations.

Mots clés: constellation; cactus; Jackson's formula; Hare-Zagier formula; maps

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Alejandro H. Morales. Bijections and symmetries for the factorizations of the long cycle. Advances in Applied Mathematics, 2013, 50, pp.702-722. ⟨hal-00653670v3⟩