Variations on shuffle squares

TL;DR

This paper explores the properties and conjectures related to shuffle squares, a class of words with symmetrical decompositions, providing new results for binary words and proposing broader conjectures for larger alphabets.

Contribution

It introduces new conjectures on generalized shuffle squares, proves that all binary words are cyclic shuffle squares, and disproves a previous conjecture about binary shuffle anti-squares.

Findings

Every binary word is a cyclic shuffle square.

Not all even binary words can be turned into shuffle squares by cyclic permutation.

Counterexamples of shuffle anti-squares exist up to length 28.

Abstract

We study decompositions of words into subwords that are in some sense similar, which means that one subword may be obtained from the other by a relatively simple transformation. Our main inspiration are shuffle squares, an intriguing class of words arising in various contexts, from purely combinatorial to more applied, like modeling concurrent processes or DNA sequencing. These words can be split into two parts that are just identical. For example, $ACTACATAGG$ is a shuffle square consisting of two copies of the word $ACTAG$. Of course, each letter must appear any even number of times in each shuffle square. We call words with that property even. We mainly discuss new problems concerning generalized shuffle squares. We propose a number of conjectures and provide some initial results towards them. We prove that every binary word is a cyclic shuffle square, meaning that it splits into two subwords, one of which is a~cyclic permutation of the other. The same statement is no longer true over larger alphabets, but it seems plausible that a similar property should hold with slightly less restricted permutation classes. For instance, we conjecture that every even ternary word is a dihedral shuffle square, which means that it splits into two subwords, one of which can be obtained from the other by a permutation corresponding to the~symmetry of a~regular polygon. We propose a general conjecture stating that a linear number of permutations is sufficient to express all even $k$-ary words as generalized shuffle squares. Our discussion is complemented by some enumerative and computational experiments. In particular, we disprove our former conjecture stating that every even binary word can be turned into a shuffle square by a cyclic permutation. The smallest counterexample has length $24$. We call words of this type shuffle anti-squares. We determined all of them up to the length $28$.