Geodesic
Exterior pairs and up step statistics on Dyck paths
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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Let $\mathcal{C}_n$ be the set of Dyck paths of length $n$. In this paper, by a new automorphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set $\mathcal{C}_n$ is equidistributed with the statistic 'number of up steps at height $h$ with $h\equiv 0$ (mod 3)'. Moreover, for $m\ge 3$, we prove that the two statistics 'number of up steps at height $h$ with $h\equiv 0$ (mod $m$)' and 'number of up steps at height $h$ with $h\equiv m-1$ (mod $m$)' on the set $\mathcal{C}_n$ are 'almost equidistributed'. Both results are proved combinatorially.
DOI : 10.37236/579
Classification : 05A15, 05A19
Mots-clés : Dyck path, exterior pair, ordered tree, planted tree, continued fraction 
@article{10_37236_579,
     author = {Sen-Peng Eu and Tung-Shan Fu},
     title = {Exterior pairs and up step statistics on {Dyck} paths},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/579},
     zbl = {1217.05023},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/579/}
}
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Sen-Peng Eu; Tung-Shan Fu. Exterior pairs and up step statistics on Dyck paths. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/579

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