On a new collection of words in the Catalan family

TL;DR

This paper introduces a new collection of words related to Catalan structures, establishing a bijection with Dyck paths and exploring connections to the zeta map and Tutte polynomial, enriching combinatorial understanding.

Contribution

It presents a novel collection of words in the Catalan family and links it to Dyck paths and algebraic structures via bijection and statistics.

Findings

Established a bijection between the new words and Dyck paths.

Connected the new collection to the zeta map in q,t-Catalan theory.

Linked the words to Tutte polynomial via planted tree statistics.

Abstract

In this note, we provide a bijection between a new collection of words on nonnegative integers of length n and Dyck paths of length 2n-2, thus proving that this collection belongs to the Catalan family. The surprising key step in this bijection is the zeta map which is an important map in the study of q,t-Catalan numbers. Finally we discuss an alternative approach to this new collection of words using two statistics on planted trees that turn out to be closely related to the Tutte polynomial on the Catalan matroid.