Roots of Binary Shuffle Squares

A square is a word of the form (Formula presented.), where X is any finite non-empty word. For example, couscous is a square. A shuffle square is a finite word that can be formed by self-shuffling a word; for instance, the Spanish word acaece is a shuffle square but not a square. We discuss both known and novel enumerative problems related to shuffle squares, with a focus on the number of distinct roots of binary shuffle squares. We introduce the term explicit shuffle squares, propose several conjectures, and present some preliminary results towards their resolution. Our discussion is supported by computational experiments. In particular, we determine the exact number of distinct roots of binary shuffle squares with a length of up to 24. On the other hand, we show that every non-constant binary word of length n generates at least n different shuffle squares.