Combinatorially refine a Zagier-Stanley result on products of permutations

TL;DR

This paper provides a simpler combinatorial approach to refine a known formula for counting permutation pairs with specific cycle structures, building on previous algebraic and character-based methods.

Contribution

It introduces a combinatorial relation that refines Zagier and Stanley's formula for products of long cycles with given cycle-types, simplifying the counting process.

Findings

Derived a simple combinatorial relation for permutation pairs

Refined previous algebraic formulas with a more straightforward approach

Enhanced understanding of cycle-type distributions in permutation products

Abstract

In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a formula of the number of pairs of long cycles whose product has $k$ cycles independently obtained by Zagier and Stanley relying on group characters, and was previously obtained by F\'{e}ray and Vassilieva by counting some colored permutations first and then relying on some algebraic computations in the ring of symmetric functions. Our approach here is simpler and combinatorial.