Geodesic
A bijection on Dyck paths and its cycle structure
The electronic journal of combinatorics, Tome 14 (2007)
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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.
DOI : 10.37236/946
Classification : 05A15, 05A19
Mots-clés : Catalan numbers, Pascal matrix, Motzkin numbers, identities 
@article{10_37236_946,
     author = {David Callan},
     title = {A bijection on {Dyck} paths and its cycle structure},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/946},
     zbl = {1112.05006},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/946/}
}
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David Callan. A bijection on Dyck paths and its cycle structure. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/946

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