Exterior Pairs and Up Step Statistics on Dyck Paths

TL;DR

This paper establishes new combinatorial equivalences between certain statistics on Dyck paths, revealing symmetries and near-symmetries in their distributions related to heights and exterior pairs.

Contribution

It introduces a novel automorphism of ordered trees to prove equidistribution and near-equidistribution of specific Dyck path statistics, extending understanding of their combinatorial properties.

Findings

Number of exterior pairs is equidistributed with up steps at heights divisible by 3.

Up steps at heights divisible by m and m-1 are almost equidistributed for m ≥ 3.

Results are proved through combinatorial methods.

Abstract

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.