On a property discovered by Xavier Grandsart

TL;DR

This paper presents a straightforward proof of a property related to circular binary words, showing that specific differences in occurrence counts are equal, which was originally discovered by Xavier Grandsart.

Contribution

The paper provides a simple and clear proof of a property concerning occurrence differences in circular binary words, clarifying a previously known but complex result.

Findings

Differences in occurrence counts are equal for circular binary words.

The proof simplifies understanding of the property.

Supports further research in combinatorics on words.

Abstract

We give a simple proof of the property discovered by Xavier Grandsart: let $W$ be a circular binary word; then the differences in the number of occurences $|W|_{0011}-|W|_{1100}$, $ |W|_{1101}-|W|_{1011}$ , $|W|_{1010}-|W|_{0101}$ and $|W|_{0100}-|W|_{0010}$ are equal.