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

Constant 2-labelling of weighted cycles

We introduce the concept of constant 2-labelling of a weighted graph and show how it can be used to obtain periodic sphere packing. Roughly speaking, a constant 2-labelling of a weighted graph is a 2-coloring (black and white) of its vertex set which preserves the sum of the weight of black vertices under some automorphisms. In this manuscript, we study this problem on weighted complete graphs and on weighted cycles. Our results on cycles allow us to determine (r,a,b)-codes in Z^2 whenever |a-b|>4 and r>1

Mathémagie - L'art de la divination

Cet exposé est basé sur la suite des exposés Mathémagie (I, II, III) de Michel Rigo. Nous présentons ici 10 tours de magie ne nécessitant aucune habileté particulière de la part de l'apprenti magicien : des tours de cartes, des tours de divination et le célèbre tour du ``barman aveugle avec des gants de boxe''. Contrairement au magicien qui ne dévoile jamais ses secrets, ici, nous expliquons que ces tours reposent sur diverses propriétés et constructions mathématiques, comme la théorie des graphes ou la combinatoire des mots

On a conjecture about regularity and l-abelian complexity

A natural generalization of automatic sequences over an infinite alphabet is given by the notion of k-regular sequences, introduced by Allouche and Shallit in 1992. The k-regularity of a sequence provides us with structural information about how the different terms are related to each other. We show that a sequence satisfying a certain symmetry property is 2-regular. We apply this theorem to develop a general approach for studying the l-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. The computation and arguments leading to these results fit into a quite general scheme that can be used to obtain additional regularity results. This supports the conjecture that the l-abelian complexity of a $k$-automatic sequence is a k-regular sequence

Some properties of abelian return words (long abstract)

We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite

Recurrence along directions in multidimensional words

In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all s_1,...,s_d, there exists n such that each block of size (n,…,n) contains the prefix of size (s1,…,sd). We are interested in a modification of this property. Namely, we ask that for each rational direction (q_1,…,q_d), each rectangular prefix occurs along this direction in positions ℓ(q1,…,qd) with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional words satisfying this condition, and more generally, a series of four increasingly stronger conditions. In particular, we study the uniform recurrence along directions of multidimentional rotation words and of fixed points of square morphisms

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Été+kayak+laval=LoL. Qu'ont en commun les termes de cette équation?

Été + kayak + Laval = LoL … Qu’ont en commun tous les termes de cette équation? Ce sont des mots palindromes! On peut les lire dans les deux sens et on obtient le même mot. Stéphanie Schanck et Élise Vandomme nous présentent un résultat récent sur les nombres palindromes. En combinant un article de J. Cilleruelo, F. Luca & L. Baxter avec un article de A. Rajasekaran, J. Shallit & T. Smith, on obtient que tout entier est la somme de trois nombres palindromes, lorsque tous ces nombres sont écrits dans une certaine base

Constant 2-labelling of a graph

We introduce the concept of constant 2-labelling of a weighted graph and show how it can be used to obtain periodic sphere packing. Roughly speaking, a constant 2-labelling of a weighted graph is a 2-coloring (black and white) of its vertex set which preserves the sum of the weight of black vertices under some automorphisms. In this manuscript, we study this problem on weighted complete graphs and on weighted cycles. Our results on cycles allow us to determine (r,a,b)-codes in Z^2 whenever |a-b|>4 and r>1

Constant 2-labellings and an application to (r,a,b)-covering codes

We introduce the concept of constant 2-labelling of a vertex-weighted graph and show how it can be used to obtain perfect weighted coverings. Roughly speaking, a constant 2-labelling of a vertex-weighted graph is a black and white colouring of its vertex set which preserves the sum of the weights of black vertices under some automorphisms. We study constant 2-labellings on four types of vertex-weighted cycles. Our results on cycles allow us to determine (r, a, b)-codes in Z^2 whenever |a−b|>4, r>1 and we give the precise values of a and b. This is a refinement of Axenovich’s theorem proved in 2003