2010/08/22 by Rosena R. X. Du, Fu Liu, Du, Rosena R. X. +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #Topological and Geometric Data Analysis #math.AG #math.CO #msc:05A15
paper · pdf · doi:10.48550/arxiv.1008.3677
23 pages, 5 figures. To appear in Graphs and Combinatorics
arxiv created 2013/12/02 · arxiv updated 2013/12/04
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 dr-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 dr-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 dr-3, where r is the number of branch points.