Minimal factorizations of a cycle: a multivariate generating function

TL;DR

This paper generalizes the counting formula for minimal factorizations of cycles in symmetric groups to a multivariate setting, also deriving related combinatorial formulas for trees and noncrossing partitions.

Contribution

It introduces a multivariate extension of the known cycle factorization formula, linking it to new results in tree hook length formulas and noncrossing partition chains.

Findings

Multivariate formula for minimal factorizations of cycles

Multivariate analog of Postnikov's hook length formula

Refined enumeration of noncrossing partition chains

Abstract

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 $n^{k-1}$ where $n$ is the rank of the underlying symmetric group and $k$ is the number of factors. In particular, this is $n^{n-2}$ 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.