Exterior Pairs and Up Step Statistics on Dyck Paths

Abstract

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.