Further results on staircase graph words

TL;DR

This paper extends the concept of staircase words by allowing larger letter differences and derives their generating functions using Chebyshev polynomials, broadening understanding of these combinatorial objects.

Contribution

It introduces a generalization of staircase words with larger letter differences and provides explicit generating functions involving Chebyshev polynomials.

Findings

Derived generating functions for generalized staircase words

Connected staircase words to Chebyshev polynomials

Extended the combinatorial framework for staircase words

Abstract

Staircase words are words in which consecutive letters do not differ by more than $1$. We generalize this by extending the restriction to letters lying further apart from each other and obtain the corresponding generating functions, which we express in terms of the Chebyshev polynomials of the second kind.