Smooth words and Chebyshev polynomials
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, and Helmut, Prodinger

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 over the alphabet is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word is said to be {\em smooth cyclic} if it is a smooth word and in addition satisfies . We find the explicit generating functions for the number of smooth words and cyclic smooth words in , 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.
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Taxonomy
Topicssemigroups and automata theory · Advanced Combinatorial Mathematics · Cellular Automata and Applications
