Anna E. Frid

Infinite permutations of lowest maximal pattern complexity

An infinite permutation @a is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we define maximal pattern complexity p@a^*(n) for infinite permutations and show that this complexity function is ultimately constant if and only if the permutation is ultimately periodic; otherwise its maximal pattern complexity is at least n, and the value p@a^*(n)=n is reached exactly on a family of permutations constructed by Sturmian words.

On possible growths of arithmetical complexity

The arithmetical complexity of infinite words, defined by Avgustinovich, Fon-Der-Flaass and the author in 2000, is the number of words of length n which occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infinite word of length n. In this paper, we show that the orders of growth of the arithmetical complexity can behave as many sub-polynomial functions. More precisely, for each sequence u of subword complexity f u (n) and for each prime p > 3 we build a Toeplitz word on the same alphabet whose arithmetical complexity is a(n) = Θ(nf u ([log pn ])).

Sequences of low arithmetical complexity

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.

On the arithmetical complexity of Sturmian words

Using the geometric dual technique by Berstel and Pocchiola, we give a uniform O(n^3) upper bound for the arithmetical complexity of a Sturmian word. We also give explicit expressions for the arithmetical complexity of Sturmian words of slope between 1/3 and 2/3 (in particular, of the Fibonacci word). In this case, the difference between the genuine arithmetical complexity function and our upper bound is bounded, and ultimately 2-periodic. In fact, our formula is valid not only for Sturmian words but for rotation words from a wider class.

On automatic infinite permutations

An infinite permutation

On possible growths of Toeplitz languages

alpha

Simple equations on binary factorial languages

is a linear ordering of

Arithmetical Complexity of Infinite Words.

mathbb N

On palindromic factorization of words

. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.

On minimal factorizations of words as products of palindromes

We consider a new family of factorial languages whose subword complexity grows as Φ(nα), where α is the only positive root of some transcendental equation. The asymptotic growth of the complexity function of these languages is studied by discrete and analytical methods, a corollary of the Wiener-Pitt theorem inclusive. The factorial languages considered are also languages of arithmetical factors of infinite words; so, we describe a new family of infinite words with an unusual growth of arithmetical complexity.