Combinatorial proofs of totals of some statistics on Catalan words

Abstract

A Catalan word is one on the alphabet of positive integers starting with $1$ in which each subsequent letter is at most one more than its predecessor. Let $\mathcal{C}_n$ denote the set of Catalan words of length $n$. In this paper, we give combinatorial proofs of explicit formulas for the sums of several parameter values taken over all the members of $\mathcal{C}_n$. In particular, we find such proofs for the parameters tracking the number of symmetric or $\ell$-valleys, which was previously requested by Baril et al. Further, we find a combinatorial explanation of a related Catalan number identity whose proof was also requested. To carry out our arguments, we consider corresponding statistics on Dyck paths and find the cardinality of certain sets of marked Dyck paths wherein one or more of the steps is distinguished from all others.