Self-shuffling Words

Abstract

AbstractIn this paper we introduce and study a new property of infinite words which is invariant under the action of a morphism: We say an infinite word \(x\in \mathbb{A}^{\mathbb N},\) defined over a finite alphabet \(\mathbb{A}\), is self-shuffling if x admits factorizations: \(x=\prod_{i=1}^\infty U_iV_i=\prod_{i=1}^\infty U_i=\prod_{i=1}^\infty V_i\) with \(U_i,V_i \in \mathbb{A}^+.\) In other words, there exists a shuffle of x with itself which reproduces x. The morphic image of any self-shuffling word is again self-shuffling. We prove that many important and well studied words are self-shuffling: This includes the Thue-Morse word and all Sturmian words (except those of the form aC where a ∈ {0,1} and C is a characteristic Sturmian word). We further establish a number of necessary conditions for a word to be self-shuffling, and show that certain other important words (including the paper-folding word and infinite Lyndon words) are not self-shuffling. In addition to its morphic invariance, which can be used to show that one word is not the morphic image of another, this new notion has other unexpected applications: For instance, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, which characterizes pure morphic Sturmian words in the orbit of the characteristic.