Manon Stipulanti

Verified

Manon verified their affiliation via an institutional email.

Learn more

Verified

Manon verified their affiliation via an institutional email.

Learn more

• Doctor of Sciences
• FNRS Research Associate (CQ) at University of Liège

Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particu...

Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{i\ge 0}$ of integers such that the value of every wo...

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT–WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of su...

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which gives the number of different factors (contiguous subblocks of their symbols), as well as abelian complexity,...

Firstly studied by Kempa and Prezza in 2018 as the unifying idea behind text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we focus on string at...

The correlation measure is a testimony of the pseudorandomness of a sequence $\infw{s}$ and provides information about the independence of some parts of $\infw{s}$ and their shifts. Combined with the well-distribution measure, a sequence possesses good pseudorandomness properties if both measures are relatively small. In combinatorics on words, the...

In combinatorics on words, the well-studied factor complexity function $\rho_{\bf x}$ of a sequence ${\bf x}$ over a finite alphabet counts, for any nonnegative integer $n$, the number of distinct length-$n$ factors of ${\bf x}$. In this paper, we introduce the \emph{reflection complexity} function $r_{\bf x}$ to enumerate the factors occurring in...

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of su...

We revisit and generalize inequalities for the summatory function of the sum of digits in a given integer base. We prove that several known results can be deduced from a theorem in a 2023 paper by Mohanty, Greenbury, Sarkany, Narayanan, Dingle, Ahnert, and Louis, whose primary scope is the maximum mutational robustness in genotype-phenotype maps.

Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for a...

Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the siz...

Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we obtain string attractors of p...

Parikh-collinear morphisms have recently received a lot of attention. They are defined by the property that the Parikh vectors of the images of letters are collinear. We first show that any fixed point of such a morphism is automatic. Consequently, we get under some mild technical assumption that the abelian complexity of a binary fixed point of a...

In formal languages and automata theory, the magic number problem can be formulated as follows: for a given integer n, is it possible to find a number d in the range [n,2n] such that there is no minimal deterministic finite automaton with d states that can be simulated by an optimal nondeterministic finite automaton with exactly n states? If such a...

Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we obtain string attractors of p...

We consider the sequence of integers whose nth term has base-p expansion given by the nth row of Pascal’s triangle modulo p (where p is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane’s On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as...

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having per...

Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for a...

Inspired by questions raised by Lejeune, we study the relationships between the k and \((k+1)\)-binomial complexities of an infinite word; as well as the link with the usual factor complexity. We show that pure morphic words obtained by iterating a Parikh-collinear morphism, i.e., a morphism mapping all words to words with bounded abelian complexit...

The notion of b-regular sequences was extended to linear recurring bases by Allouche, Scheicher and Tichy in 2000, and to abstract numeration systems by Maes and Rigo in 2002. Their definitions are based on a notion of S-kernel that extends that of b-kernel. However, these definitions do not allow us to generalize all of the many characterizations...

Regular sequences generalize the extensively studied automatic sequences. Let S be an abstract numeration system. When the numeration language L is prefix-closed and regular, a sequence is said to be S-regular if the module generated by its S-kernel is finitely generated. In this paper, we give a new characterization of such sequences in terms of t...

A word w is said to be closed if it has a proper factor x which occurs exactly twice in w, as a prefix and as a suffix of w. Based on the concept of Ziv-Lempel factorization, we define the closed z-factorization of finite and infinite words. Then we find the closed z-factorization of the infinite m-bonacci words for all m≥2. We also classify closed...

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base $\beta$. However, the given condition turns to be not necessary. Hence, the goal of this paper is to provide...

We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears nat...

Two words are $k$-binomially equivalent, if each word of length at most $k$ occurs as a subword, or scattered factor, the same number of times in both words. The $k$-binomial complexity of an infinite word maps the natural $n$ to the number of $k$-binomial equivalence classes represented by its factors of length $n$. Inspired by questions raised by...

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base \(\beta \). However, the given condition turns out to be not necessary. Hence, the goal of this paper is to p...

A word $w$ is said to be closed if it has a proper factor $x$ which occurs exactly twice in $w$, as a prefix and as a suffix of $w$. Based on the concept of Ziv-Lempel factorization, we define the closed $z$-factorization of finite and infinite words. Then we find the closed $z$-factorization of the infinite $m$-bonacci words for all $m \geq 2$. We...

We consider numeration systems based on a d-tuple U=(U1,…,Ud) of sequences of integers and we define (U,K)-regular sequences through K-recognizable formal series, where K is any semiring. We show that, for any d-tuple U of Pisot numeration systems and any semiring K, this definition does not depend on the greediness of the U-representations of inte...

Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated by its $S$-kernel is finitely generated. In this paper, we give a new characterization of such sequences in te...

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having per...

The notion of $b$-regular sequences was generalized to abstract numeration systems by Maes and Rigo in 2002. Their definition is based on a notion of $\mathcal{S}$-kernel that extends that of $b$-kernel. However, this definition does not allow us to generalize all of the many characterizations of $b$-regular sequences. In this paper, we present an...

We identify the structure of the lexicographically least word avoiding $5/4$-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $\mathbf{p} \, \tau ( \varphi(\mathbf{z}) \varphi^2(\mathbf{z}) \cdots)$ where $\mathbf{p},\mathbf{z}$ are finite words, $\varphi$ is a $6$-uniform morphism, and $\tau$ is a c...

We consider numeration systems based on a $d$-tuple $\mathbf{U}=(U_1,\ldots,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any $d$-tuple $\mathbf{U}$ of Pisot numeration systems and any commutative semirin...

We identify the structure of the lexicographically least word avoiding 5/4-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $p \tau(\varphi(z) \varphi^2(z) \cdots)$ where $p, z$ are finite words, $\varphi$ is a 6-uniform morphism, and $\tau$ is a coding. This description yields a recurrence for the $...

We identify the structure of the lexicographically least word avoiding 5/4-powers on the alphabet of nonnegative integers.

We make certain bounds in Krebs’ proof of Cobham’s theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic a-automatic sequence and an aperiodic b-automatic sequence, where a and b are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors in common...

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which...

We introduce a variation of the Ziv-Lempel and Crochemore factorizations of words by requiring each factor to be a palindrome. We compute these factorizations for the Fibonacci word, and more generally, for all $m$-bonacci words.

We introduce a variation of the Ziv–Lempel and Crochemore factorizations of words by requiring each factor to be a palindrome. We compute these factorizations for the Fibonacci word, and more generally, for all m-bonacci words.

We make certain bounds in Krebs' proof of Cobham's theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic $a$-automatic sequence and an aperiodic $b$-automatic sequence, where $a$ and $b$ are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors i...

If $p$ is a prime number, consider a $p$-automatic sequence $(u_n)_{n\ge 0}$, and let $U(X) = \sum_{n\ge 0} u_n X^n \in \mathbb{F}_p[[X]]$ be its generating function. Assume that there exists a formal power series $V(X) = \sum_{n\ge 0} v_n X^n \in \mathbb{F}_p[[X]]$ which is the compositional inverse of $U$, i.e., $U(V(X))=X=V(U(X))$. The problem i...

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which...

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand propert...

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand propert...

If p is a prime number, consider a p-automatic sequence (un)n≥0, and let U(X) =∑n≥0 unXⁿ ∈ Fp [[X]] be∑ its generating function. Assume that there exists a formal power series V(X) =n≥0 vnXⁿ ∈ Fp [[X]] which is the compositional inverse of U, i.e., U(V(X)) = X = V(U(X)). The problem investigated in this paper is to study the properties of the seque...

We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a generalization of the Pascal triangle to binomial coefficients of base-$b$ expansions. By using a convenient tree s...

Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply...

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a r...

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpi\'nski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact bloc...

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence (S(n))n≥0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrenc...

Many digital functions studied in the literature, e.g., the summatory function of the base-k sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply i...

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks...