Breaking Cycles, the Odd Versus the Even

TL;DR

This paper explores the relationship between permutations with odd and even cycles, connecting various combinatorial structures through a recursive cycle-breaking approach, revealing deep links in permutation theory.

Contribution

It introduces an intermediate combinatorial structure linking odd and even cycle permutations, unifying previous results with a novel recursive cycle-breaking method.

Findings

Established a bijection connecting odd and even cycle permutations.

Revealed a recursive cycle-breaking technique as a key tool.

Linked different permutation classes through a unified framework.

Abstract

In an award-winning expository article, V. Pozdnyakov and J.M. Steele gave a beautiful demonstration of the ramifications of a basic bijection for permutations. The aim of this note is to connect this correspondence to a seemingly unrelated problem concerning odd cycles and even cycles, arising in the combinatorial study of the Cayley continuants by E. Munarini and D. Torri. In extreme cases, one encounters two special classes of permutations of $2n$ elements with the same cardinality. A bijection of this appealing relation has been found by E. Sayag. A combinatorial study of permutations with only odd cycles has been carried out by M. B\'ona, A. Mclennan and D. White. We find an intermediate structure which leads to a linkage between these two antipodal structures. A recursive setting reveals that everything boils down to only one trick -- breaking the cycles.