Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k >= 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = infinity). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: Each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

Variations of the Morse-Hedlund theorem for k-abelian equivalence

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ≥ 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = ∞). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

On cardinalities of k-abelian equivalence classes

Two words $u$ and $v$ are $k$-abelian equivalent if for each word $x$ of length at most $k$, $x$ occurs equallymany times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the $k$-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton $k$-abelian classes, i.e., classes containing only one element. We find a connection between thesingleton classes and cycle decompositions of the de Bruijn graph. We show that the number of classes of words of length $n$ containing one single element is of order $mathcal O (n^{N_m(k-1)-1})$, where $N_m(l)= frac{1}{l}sum_{dmid l}arphi(d)m^{l/d}$ is the number of necklaces of length $l$ over an $m$-ary alphabet. We conjecture that the upper bound is sharp. We also remark that, for $k$ even and $m=2$, the lower bound $Omega (n^{N_m(k-1)-1})$follows from an old conjecture on the existence of Gray codes for necklaces of odd length. We verify this conjecture for necklaces of length up to 15

A new approach to the 22-regularity of the ℓ\ell-abelian complexity of 22-automatic sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.Comment: 44 pages, 2 figures; publication versio

Templates for the k-binomial complexity of the Tribonacci word

Consider k-binomial equivalence: two finite words are equivalent if they share the same subwords of length at most k with the same multiplicities. With this relation, the k-binomial complexity of an infinite word x maps the integer n to the number of pairwise non-equivalent factors of length n occurring in x. In this paper based on the notion of template introduced by Currie et al., we show that, for all k≥2, the k-binomial complexity of the Tribonacci word coincides with its usual factor complexity p(n)=2n+1. A similar result was already known for Sturmian words, but the proof relies on completely different techniques that seemingly could not be applied for Tribonacci. © 2019 Elsevier Inc