arXiv subjects

Marie Lejeune

Publications and source records attributed to Marie Lejeune.

4 recordsLinked to original sources

arXiv2021

$\mathcal{S}$-adic characterization of minimal ternary dendric shifts

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal{S}$-adic representation where the morphisms in $\mathcal{S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper we investigate those $\mathcal{S}$-adic representations, heading towards an $\mathcal{S}$-adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with 2 vertices.

math.DS↗

arXiv2020

Reconstructing Words from Right-Bounded-Block Words

A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word $w \in \{a, b\}^{*}$ can be reconstructed from the number of occurrences of at most $\min(|w|_a, |w|_b)+ 1$ scattered factors of the form $a^{i} b$. Moreover, we generalize the result to alphabets of the form $\{1,\ldots,q\}$ by showing that at most $ \sum^{q-1}_{i=1} |w|_i (q-i+1)$ scattered factors suffices to reconstruct $w$. Both results improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here.

cs.FL↗

arXiv2020

On the binomial equivalence classes of finite words

Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word $x$ of length at most $k$, $x$ appears the same number of times as a subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the $k$-binomial equivalence with a special focus on the cardinalities of the classes. We provide an algorithm generating the $2$-binomial equivalence class of a word. For $k \geq 2$ and alphabet of $3$ or more symbols, the language made of lexicographically least elements of every $k$-binomial equivalence class and the language of singletons, i.e., the words whose $k$-binomial equivalence class is restricted to a single element, are shown to be non context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-$2$ group on $m$ generators is isomorphic to the quotient of the free monoid $\{ 1, \ldots , m\}^{*}$ by the $2$-binomial equivalence.

cs.FL↗

arXiv2018

Computing the $k$-binomial complexity of the Thue--Morse word

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $Ψ^\ell(w)$ for an arbitrary morphism $Ψ$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation.

cs.DM↗