Chebyshev Polynomials and Statistics on a New Collection of Words in the Catalan Family

TL;DR

This paper introduces a novel method using Chebyshev polynomials to analyze statistics on a new class of words related to Catalan numbers, providing explicit formulas and generating functions.

Contribution

It develops a new approach employing Chebyshev polynomials to derive explicit formulas and generating functions for statistics on the word class L_n, extending previous results.

Findings

Explicit formulas for zeros and ones in L_n

Functional equations solved via Chebyshev polynomials

Recurrences and generating functions for general i

Abstract

Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of occurrences of a letter i. In the cases i = 0 and i = 1, we are able to determine explicit expressions for the number of members of L_n containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employing Chebyshev polynomials. Recurrences and generating function formulas are also provided in the case of general i.