Tree-like properties of cycle factorizations

Abstract

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 Dénes 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.

arXiv comment10 pages, 3 figures