Minimal factorizations of a cycle: a multivariate generating function

Philippe Biane (1) , Matthieu Josuat-Vergès (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

Domaines: Combinatoire [math.CO]

Philippe Biane, Matthieu Josuat-Vergès. Minimal factorizations of a cycle: a multivariate generating function. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6318⟩. ⟨hal-02173738⟩