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Generalizations of Parikh mappings

Anton Černý (2010)

RAIRO - Theoretical Informatics and Applications

Parikh matrices have become a useful tool for investigation of subword structure of words. Several generalizations of this concept have been considered. Based on the concept of formal power series, we describe a general framework covering most of these generalizations. In addition, we provide a new characterization of binary amiable words – words having a common Parikh matrix.

Generalized golden ratios of ternary alphabets

Vilmos Komornik, Anna Chiara Lai, Marco Pedicini (2011)

Journal of the European Mathematical Society

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence...

Generalized Thue-Morse words and palindromic richness

Štěpán Starosta (2012)

Kybernetika

We prove that the generalized Thue-Morse word 𝐭 b , m defined for b 2 and m 1 as 𝐭 b , m = s b ( n ) mod m n = 0 + , where s b ( n ) denotes the sum of digits in the base- b representation of the integer n , has its language closed under all elements of a group D m isomorphic to the dihedral group of order 2 m consisting of morphisms and antimorphisms. Considering antimorphisms Θ D m , we show that 𝐭 b , m is saturated by Θ -palindromes up to the highest possible level. Using the generalisation of palindromic richness recently introduced by the author and E. Pelantová,...

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