Exterior Pairs and Up Step Statistics on Dyck Paths

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 $hequiv 0$ (mod 3). Moreover, for $mge 3$, we prove that the two statistics `number of up steps at height $h$ with $hequiv 0$ (mod $m$) and `number of up steps at height $h$ with $hequiv m-1$ (mod $m$) on the set $C_n$ are `almost equidistributed. Both results are proved combinatorially.