Factorizations of Cycles and Multi-Noded Rooted Trees

Abstract

In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a d-cycle into a product of cycles of prescribed lengths is d r-2. To prove our result, we first define a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the cardinality of this new class which with proper parameters is exactly d r-2. The main part of this paper is the proof that there is a bijection from factorizations of a d-cycle to multi-noded rooted trees via factorization graphs. This implies the desired formula. The factorization problem we consider has its origin in geometry, and is related to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the standard translation of Hurwitz numbers into group theory, our main result is equivalent to the following: when the genus is 0 and one of the ramification indices is d, the degree of the covers, the pure-cycle Hurwitz number is d r-3, where r is the number of branch points.