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