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.
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R. R. X. Du is partially supported by the National Science Foundation of China under Grant No. 10801053, Shanghai Rising-Star Program (No. 10QA1401900), and the Fundamental Research Funds for the Central Universities. F. Liu is partially supported by the National Security Agency under Grant No. H98230-09-1-0029, and the National Science Foundation of China under Grant No. 10801053.
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Du, R.R.X., Liu, F. Factorizations of Cycles and Multi-Noded Rooted Trees. Graphs and Combinatorics 31, 551–575 (2015). https://doi.org/10.1007/s00373-013-1404-y
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DOI: https://doi.org/10.1007/s00373-013-1404-y