On a property discovered by Xavier Grandsart
  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. 
Pierre Arnoux
math.CO