Note on occurrences of factors in circular words

Abstract

We give an elementary proof of a property discovered by Xavier Grandsart: let W be a circular binary word; then the differences in the number of occurrences |W |0011 − |W |1100 , |W |1101 − |W |1011 , |W |1010 − |W |0101 and |W |0100 − |W |0010 are equal; this property is easily generalized using the De Bruijn graph.