Breaking Cycles, the Odd Versus the Even

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.