Shuffle squares and ordered nest-free graphs

Abstract

A shuffle square is a word consisting of two shuffled copies of the same word. For instance, the Turkish word $\mathtt{\color{red}{ik}\color{blue}{i}\color{red}{li}\color{blue}{kli}}$ (binary in English) is a shuffle square, as it can be split into two copies of the word $\mathtt{ikli}$. We explore a representation of shuffle squares in terms of \emph{ordered nest-free graphs} and demonstrate the usefulness of this approach by applying it to several families of binary words. Among others, we characterize shuffle squares with four and five runs, as well as shuffle squares with all $\mathtt1$-runs of length one (and with the $\mathtt1$'s alternating between the two copies). In our main result we provide quite general sufficient conditions for a binary word not to be a shuffle square. In particular, it follows that binary words of the type $(\mathtt{1001})^n$, $n$ odd, are not shuffle squares. We complement it by showing that all other words whose every $\mathtt{1}$-run has length one or two, while every $\mathtt{0}$-run has length two, are shuffle squares. We also provide a counterexample to a believable stipulation that binary words of the form $\mathtt1^{m}\mathtt0^{m-2}\mathtt1^{m-4}\cdots$, $m$ odd, are far from being shuffle squares (the distance measured by the minimum number of letters one has to delete in order to turn a word into a shuffle square).