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

La Faculté des Sciences

Ce texte retrace, clans ses grandes lignes, l’histoire de la Faculté des Sciences de Lille, de 1854 à 1970. On ne trouvera pas ici l’histoire détaillée de disciplines, qui fait l’objet de contributions séparées. LA CRÉATION DE LA FACULTÉ Après les journées de juin 1848 et les élections, l’Assemblée et le gouvernement cherchent à limiter l’influence des enseignants d’État (on est dans le régime du monopole depuis Napoléon) et de renforcer l’influence de l’Église. La loi Falloux..

A conjecture on the 2-abelian complexity of the Thue-Morse word

The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. For example, it is trivially abelian periodic and its abelian complexity takes only two values. For an integer k, the k-abelian complexity is a generalization of the abelian complexity, corresponding to the case where k=1. Formally, two words u and v of the same length are k-abelian equivalent if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of occurrences of x in u and v are the same. This notion has received some recent interest, see the works of Karhumäki et al. The k-abelian complexity of an infinite word x maps an integer n to the number of k-abelian classes partitioning the set of factors of length n occurring in x. The aim of this talk is to study the 2-abelian complexity a(n) of the Thue-Morse word. We conjecture that a(n) is 2-regular in the sense of Allouche and Shallit. This question can be related to a work of Madill and Rampersad (2012) where the (1)-abelian complexity of the paper folding word is shown to be 2-regular. We will present some arguments supporting our conjecture. They are based on functions counting some subword of length 2 occuring in prefixes of the Thue-Morse word

A New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-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 Z-module generated by its 2-kernel is finitely generated. 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. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular

A new approach to the 2-regularity of the ℓ-abelian complexity of 2-automatic sequences (extended abstract)

We show that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit. 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. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular

Deciding game invariance

International audienceDuchêne and Rigo introduced the notion of invariance for take-away games on heaps. Roughly speaking, these are games whose rulesets do not depend on the position. Given a sequence S of positive tuples of integers, the question of whether there exists an invariant game having S as set of P-positions is relevant. In particular, it was recently proved by Larsson et al. that if $S$ is a pair of complementary Beatty sequences, then the answer to this question is always positive. In this paper, we show that for a fairly large set of sequences (expressed by infinite words), the answer to this question is decidable