A bijection on Dyck paths and its cycle structure

Details A bijection on Dyck paths and its cycle structure A bijection on Dyck paths and its cycle structure

2006-11-22

arxiv.org/abs/math/0611698v1

Mathematics

Combinatorics

17 pages. Uses PSTricks for tree diagrams and Krattenthaler's LaTeX code for lattice path diagrams. No external (.eps) figures

Scientific paper

The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each cycle has length a power of 2. A new manifestation of the Catalan numbers as labeled forests crops up enroute as does the Pascal matrix mod 2. We use the bijection to show the equivalence of two known manifestations of the Motzkin numbers. Finally, we consider some statistics on the new Catalan manifestation.

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