Exterior Pairs and Up Step Statistics on Dyck Paths

Let ${\text{C}}_n$ 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 ${\text{C}}_n$ are `almost equidistributed’. Both results are proved combinatorially.