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 ?n whose product is (12...n), and labelled trees on n vertices. We prove a refinement of a theorem of J. Dénes (1959, Publ. Math. Inst. Hungar. Acad. Sci. 4, 63-71) that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization corresponds naturally under our bijection to leaf edges (incident with a vertex of degree 1) of a tree. Moreover, we give a generalization of this fact. © 2002 Elsevier Science (USA).