Some Results on Digital Segments and Balanced Words

TL;DR

This paper explores combinatorial properties of Christoffel and balanced words, providing formulas for counting balanced words and analyzing minimal non-balanced words, motivated by their geometric interpretation as digital segments.

Contribution

It introduces a closed formula for counting balanced words with given zeroes and ones and studies minimal non-balanced words, advancing understanding of their combinatorial structure.

Findings

Derived a closed formula for the number of balanced words with specified zeroes and ones

Analyzed properties of minimal non-balanced words

Connected combinatorial results to geometric interpretations of digital segments

Abstract

We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with $a$ zeroes and $b$ ones. We also study minimal non-balanced words.