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COMBINATORICS
2007

A Bijection on Dyck Paths and its Cycle Structure

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A Bijection on Dyck Paths and its Cycle Structure
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 en route 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 and the identities they interpret.
David Callan
Added 12 Dec 2010
Updated 12 Dec 2010
Type Journal
Year 2007
Where COMBINATORICS
Authors David Callan
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