Tree-like properties of cycle factorizations

Details Tree-like properties of cycle factorizations Tree-like properties of cycle factorizations

2001-06-06

arxiv.org/abs/math/0106039v1

J. Combinatorial Theory ser. A, 98 (2002), 106--117

Mathematics

Combinatorics

10 pages, 3 figures

Scientific paper

We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a theorem of D\'{e}nes that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization correspond naturally under our bijection to leaf edges of a tree. Moreover, we give a generalization of this fact.

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