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Manon Stipulanti

Publications and source records attributed to Manon Stipulanti.

At least 19 recordsLinked to original sources
arXiv2026

Symbols frequencies in the Thue--Morse word in base $3/2$ and related conjectures

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms rather than a single primitive substitution, classical Perron--Frobenius methods do not directly apply to determine symbol frequencies. We prove that both symbols ${\tt 0},{\tt 1}$ occur with frequency $1/2$ and we show uniform recurrence and symmetry properties of its set of factors. The proof reveals a structural bridge between combinatorics on words and harmonic analysis: the first difference sequence is shown to be Toeplitz, providing dynamical rigidity, while filtered frequencies naturally encode a dyadic structure that lifts to the compact group of $2$-adic integers. In this $2$-adic setting, desubstitution becomes a linear operator on Fourier coefficients, and a spectral contraction argument enforces uniqueness of limiting densities. Our results answer several conjectures of Dekking (on a sibling sequence) and illustrate how harmonic analysis on compact groups can be fruitfully combined with substitution dynamics.

math.CO↗
arXiv2026

Factor-balancedness, linear recurrence, and factor complexity

In the study of infinite words, various notions of balancedness provide quantitative measures for how regularly letters or factors occur, and they find applications in several areas of mathematics and theoretical computer science. In this paper, we study factor-balancedness and uniform factor-balancedness, making two main contributions. First, we establish general sufficient conditions for an infinite word to be (uniformly) factor-balanced, applicable in particular to any given linearly recurrent word. These conditions are formulated in terms of $\mathcal{S}$-adic representations and generalize results of Adamczewski on primitive substitutive words, which show that balancedness of length-2 factors already implies uniform factor-balancedness. As an application of our criteria, we characterize the Sturmian words and ternary Arnoux--Rauzy words that are uniformly factor-balanced as precisely those with bounded weak partial quotients. Our second main contribution is a study of the relationship between factor-balancedness and factor complexity. In particular, we analyze the non-primitive substitutive case and construct an example of a factor-balanced word with exponential factor complexity, thereby making progress on a question raised in 2025 by Arnoux, Berth\'e, Minervino, Steiner, and Thuswaldner on the relation between balancedness and discrete spectrum.

math.CO↗
arXiv2025

Doubling modulo odd integers, generalizations, and unexpected occurrences

The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive integer $n$. More precisely, this doubling map defines a permutation $\sigma_{2,n}$ and each of $A$ and $B$ counts the number $C_2(n)$ of cycles of $\sigma_{2,n}$, hence $A=B$. In the first part of this note, we give a direct proof of this last equality. To do so, we consider and study a generalized $(k,n)$-perfect shuffle permutation $\sigma_{k,n}$, where we multiply by an integer $k\ge 2$ instead of $2$, and its number $C_k(n)$ of cycles. The second part of this note lists some of the many occurrences and applications of the doubling map and its generalizations in the literature: in mathematics (combinatorics of words, dynamical systems, number theory, correcting algorithms), but also in card-shuffling, juggling, bell-ringing, poetry, and music composition.

math.NT↗
arXiv2025

Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this notion? When can these equivalence relations and complexity functions be computed efficiently? We study the fixed points of substitution of Pisot type. Each of their $k$-abelian complexities is bounded and the Parikh vectors of their length-$n$ prefixes form synchronized sequences in the associated Dumont--Thomas numeration system. Therefore, the $k$-abelian complexity of Pisot substitution fixed points is automatic in the same numeration system. Two effective generic construction approaches are investigated using the \texttt{Walnut} theorem prover and are applied to several examples. We obtain new properties of the Tribonacci sequence, such as a uniform bound for its factor balancedness together with a two-dimensional linear representation of its generalized abelian complexity functions.

cs.FL↗
arXiv2025

Positionality of Dumont--Thomas numeration systems for integers

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 word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by R\'{e}nyi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995

math.CO↗
arXiv2024

Additive word complexity and Walnut

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, which counts factors up to a permutation of letters. In this paper, we consider the relatively unexplored concept of additive complexity, which counts the number of factors up to additive equivalence. We say that two words are additively equivalent if they have the same length and the total weight of their letters is equal. Our contribution is to expand the general knowledge of additive complexity from a theoretical point of view and consider various famous examples. We show a particular case of an analog of the long-standing conjecture on the regularity of the abelian complexity of an automatic sequence. In particular, we use the formalism of logic, and the software Walnut, to decide related properties of automatic sequences. We compare the behaviors of additive and abelian complexities, and we also consider the notion of abelian and additive powers. Along the way, we present some open questions and conjectures for future work.

math.CO↗
arXiv2024

On the pseudorandomness of Parry--Bertrand automatic sequences

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 famous $b$-automatic sequences are quite far from being pseudorandom, as they have small factor complexity on the one hand and large well-distribution and correlation measures on the other. This paper investigates the pseudorandomness of a specific family of morphic sequences, including classical $b$-automatic sequences. In particular, we show that such sequences have large even-order correlation measures; hence, they are not pseudorandom. We also show that even- and odd-order correlation measures behave differently when considering some simple morphic sequences.

math.CO↗
arXiv2024

The reflection complexity of sequences over finite alphabets

In combinatorics on words, the well-studied factor complexity function $\rho_{\infw{x}}$ of a sequence $\infw{x}$ over a finite alphabet counts, for every nonnegative integer $n$, the number of distinct length-$n$ factors of $\infw{x}$. In this paper, we introduce the \emph{reflection complexity} function $r_{\infw{x}}$ to enumerate the factors occurring in a sequence $\infw{x}$, up to reversing the order of symbols in a word. We prove a number of results about the growth properties of $r_{\infw{x}}$ and its relationship with other complexity functions. We also prove a Morse--Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. We investigate the reflection complexity of quasi-Sturmian, episturmian, $(s+1)$-dimensional billiard, complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if $\infw{x}$ is $k$-automatic, then $r_{\infw{x}}$ is computably $k$-regular, and we use the software \texttt{Walnut} to evaluate the reflection complexity of some automatic sequences, such as the Thue--Morse sequence. We note that there are still many unanswered questions about this reflection measure.

math.CO↗
arXiv2024

Automatic Abelian Complexities of Parikh-Collinear Fixed Points

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 such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.

cs.DM↗
arXiv2024

Combinatorics on words and generating Dirichlet series of automatic sequences

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 particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base $b$; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet $\{0, 1, \dots, b-1\}$ that do not begin with a $0$. We show how to unify and extend results proved by Nathanson in 2021 and by K\"ohler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous $b$-automatic sequences or $b$-regular sequences. We also consider a specific sequence that is not $b$-regular.

math.CO↗
arXiv2023

Summing the sum of digits

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.

math.NT↗
arXiv2023

An elementary proof of Bridy's theorem

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 size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions.

math.NT↗
arXiv2023

Magic numbers in periodic sequences

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,2^n] 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 number d exists, it is called magic. In this paper, we consider the magic number problem in the framework of deterministic automata with output, which are known to characterize automatic sequences. More precisely, we investigate magic numbers for periodic sequences viewed as either automatic, regular, or constant-recursive.

cs.FL↗
arXiv2023

String attractors of some simple-Parry automatic sequences

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 prefixes of particular infinite words generalizing k-bonacci words (including the famous Fibonacci word) and related to simple Parry numbers. In fact, our description involves the numeration systems classically derived from the considered morphisms. This extends our previous work published in the international conference WORDS 2023.

math.CO↗
arXiv2022

On extended boundary sequences of morphic and Sturmian words

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 all pairs of factors of length $\ell$ separated by $n-\ell$ symbols. For the large class of addable abstract numeration systems $S$, we show that if an infinite word is $S$-automatic, then the same holds for its $\ell$-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and $S$-automatic words for which the boundary sequence is nevertheless $S$-automatic and conversely, $S$-automatic words with a boundary sequence that is not $S$-automatic. In the second part of the paper, we study the $\ell$-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.

math.CO↗
arXiv2022

A full characterization of Bertrand numeration systems

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 a correction of Bertrand-Mathis' result. The main difference arises when $\beta$ is a Parry number, in which case are derived two associated Bertrand numeration systems. Along the way, we define a non-canonical $\beta$-shift and study its properties analogously to those of the usual canonical one.

math.CO↗
arXiv2022

On digital sequences associated with Pascal's triangle

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 naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo~$p$. This note ends with a discussion about Pascal's pyramid involving trinomial coefficients.

math.NT↗