Exterior Pairs and Up Step Statistics on Dyck Paths

Let Cn 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 Cn is equidistributed with the statistic ‘number of up steps at height h with h ≡ 0 (mod 3)’. Moreover, for m ≥ 3, we prove that the two statistics ‘number of up steps at height h with h ≡ 0 (mod m)’ and ‘number of up steps at height h with h ≡ m − 1 (mod m)’ on the set Cn are ‘almost equidistributed’. Both results are proved combinatorially.