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In an award-winning expository article, V. Pozdnyakov and J.M. Steele gave abeautiful demonstration of the ramifications of a basic bijection forpermutations. The aim of this note is to connect this correspondence to aseemingly unrelated problem concerning odd cycles and even cycles, arising inthe 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 hasbeen found by E. Sayag. A combinatorial study of permutations with only oddcycles has been carried out by M. B\'ona, A. Mclennan and D. White. We find anintermediate structure which leads to a linkage between these two antipodalstructures. A recursive setting reveals that everything boils down to only onetrick -- breaking the cycles.

Breaking Cycles, the Odd Versus the Even