Tree-like properties of cycle factorizations

Authors:
Ian Goulden;Alexander Yong
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada;Department of Mathematics, University of Michigan, Ann Arbor, Michigan
Venue:
Journal of Combinatorial Theory Series A
Year:
2002

Citing 3
Cited 3

Constructive combinatorics

Constructive combinatorics
A solution to a problem of Denes: a bijection between trees and factorizations of Cyclic permutations

European Journal of Combinatorics
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Transitive cycle factorizations and prime parking functions

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Some factorisations counted by Catalan numbers

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Cycle factorizations and 1-faced graph embeddings

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Abstract

We provide a bijection between the set of factorizations, that is, ordered (n-1)- tuples of transpositions in Jn 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.