On inequivalent factorizations of a cycle

AbstractWe introduce a bijection between inequivalent minimal factorizations ofthe n-cycle (12...n) into a product of smaller cycles of given length and treesof a certain structure. A factorization has the type α = (α 2 ,α 3 ,···) if it hasα j factors of length j. Inequivalent factorizations are defined up to reorderingof commuting factors. A factorization is minimal if no factorizations of a typeα ′ strictly smaller than α exist.The introduced bijection allows us to answer such questions as the numberof factorizations with a given number of different (commuting) factors thatcan appear in the first and in the last positions, and the structure of theset of factors that can be arranged into a product evaluating to (12...n).Important consequences of the discovered structure include monotonicity ofthe constituent factors and uniqueness of an arrangement into a valid factor-ization: any two minimal factorizations of (12...n) consisting of the samefactors must be equivalent. 1 Introduction Counting factorizations of a permutation into a product of cycles of specified lengthis a problem with rich history, dating back at least to Hurwitz [1], and with manyimportant applications, in particular in geometry (see e.g. [2]). In this article weare concerned with the