On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet

TL;DR

This paper explores the enumeration and asymptotic analysis of balanced words over a two-letter alphabet, linking them to discrete line segments and focusing on symmetric cases like palindromes.

Contribution

It introduces a recurrence relation for counting balanced words of given length and height, and provides generating functions and asymptotic results for these counts.

Findings

Derived a recurrence for counting balanced words

Provided generating functions for these counts

Analyzed asymptotic behavior of the counts

Abstract

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.