A versatile combinatorial approach of studying products of long cycles in symmetric groups

TL;DR

This paper develops a combinatorial framework to enumerate pairs of long cycles in symmetric groups with prescribed properties, unifying and extending previous results, and providing new explicit formulas for permutation factorizations and separation probabilities.

Contribution

It introduces a versatile combinatorial approach to study products of long cycles, deriving general formulas and solving open problems in permutation factorizations and separation probabilities.

Findings

Unified formulas for long cycle products and cycle-type separation

Explicit enumeration of long cycle pairs respecting set partitions

New formulas for factorizations of even permutations and separation probabilities

Abstract

In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial proofs, including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new results as well. In particular, for the first time, given a partition of a set, we obtain an explicit formula for the number of pairs of long cycles on the set such that the product of the long cycles does not mix the elements from distinct blocks of the partition and has an independently prescribed number of cycles for each block of elements. As applications, we obtain new explicit formulas concerning factorizations of any even permutation into long cycles and the first nontrivial explicit formula for computing strong separation probabilities solving an open problem of Stanley (2010).