Exterior Pairs and Up Step Statistics on Dyck Paths

Details Exterior Pairs and Up Step Statistics on Dyck Paths Exterior Pairs and Up Step Statistics on Dyck Paths

2010-10-27

arxiv.org/abs/1010.5673v1

Mathematics

Combinatorics

13 pages

Scientific paper

Let $\C_n$ 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 $\C_n$ 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 $\C_n$ are `almost equidistributed'. Both results are proved combinatorially.

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