Smooth words and Chebyshev polynomials

TL;DR

This paper derives explicit generating functions and formulas for counting smooth and cyclic smooth words over finite alphabets using Chebyshev polynomials, providing asymptotic analysis and enumeration of related necklaces.

Contribution

It introduces explicit generating functions and formulas for smooth words and cyclic smooth words, connecting combinatorial enumeration with Chebyshev polynomials and trigonometric sums.

Findings

Explicit generating functions in terms of Chebyshev polynomials

Closed-form formulas as trigonometric sums

Asymptotic estimates for counts of smooth words

Abstract

A word $\sigma=\sigma_1...\sigma_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $\sigma$ is said to be {\em smooth cyclic} if it is a smooth word and in addition satisfies $|\sigma_n-\sigma_1|\le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {\it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation.