Combinatorial proofs of totals of some statistics on Catalan words

TL;DR

This paper provides combinatorial proofs for formulas related to sums of parameters on Catalan words, including symmetric and valley counts, and explains a Catalan number identity through Dyck path analysis.

Contribution

It introduces new combinatorial proofs for parameter sums on Catalan words and offers explanations for Catalan number identities using Dyck path statistics.

Findings

Explicit formulas for sums of parameters on Catalan words

Combinatorial proofs for symmetric and valley parameters

A new explanation for a Catalan number identity

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.