Counting pairs of cycles whose product is a permutation with restricted cycle lengths

TL;DR

This paper derives formulas for counting pairs of cycles with restricted lengths whose product is a permutation, and applies these results to establish a lower bound on permutations with high block transposition distance.

Contribution

It provides exact and asymptotic formulas for counting specific cycle pairs and connects these to permutation distance bounds, a novel combination of combinatorics and permutation analysis.

Findings

Formulas for the number of cycle pairs with restricted lengths

Asymptotic estimates for large N

A new lower bound for permutation block transposition distance

Abstract

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower bound for the number of permutations whose block transposition distance from the identity is at least $(n+1)/2$.