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Combinatorics on words and generating Dirichlet series of automatic sequences
Allouche, Jean-Paul; Shallit Jeffrey; Stipulanti, Manon


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Keywords :
Combinatorics on words; Generating Dirichlet series; Automatic sequences; Missing digits; Restricted words
Abstract :
[en] 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 on 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 $q$-automatic sequences or $q$-regular sequences.
Disciplines :
Author, co-author :
Allouche, Jean-Paul
Shallit Jeffrey
Stipulanti, Manon  ;  Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
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Title :
Combinatorics on words and generating Dirichlet series of automatic sequences
Publication date :
Number of pages :
Funders :
F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]
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since 11 December 2023


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