On inequivalent factorizations of a cycle

We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 . . . n) into a product of smaller cycles of given length and trees of a certain structure. A factorization has the type α = (α2, α3, · · · ) if it has αj factors of length j. Inequivalent factorizations are defined up to reordering of commuting factors. A factorization is minimal if no factorizations of a type α ′ strictly smaller than α exist. The introduced bijection allows us to answer such questions as the number of factorizations with a given number of different (commuting) factors that can appear in the first and in the last positions, and the structure of the set of factors that can be arranged into a product evaluating to (1 2 . . . n). Important consequences of the discovered structure include monotonicity of the constituent factors and uniqueness of an arrangement into a valid factorization: any two minimal factorizations of (1 2 . . . n) consisting of the same factors must be equivalent.