On automatic infinite permutations

Abstract

An infinite permutation α 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 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.Une permutation infinie est un ordre total sur N. Nousétudions les caractéristiques des permutations infinies qui sont analoguesà celles des mots infinis, et nous montrons que certaines d'entre elles se comportent de la même façon, et d'autres pas. Dans cet article nous essayons d'étendre la notion d'automaticité de mots aux permutations. Cependant il arrive que des définitionséquivalentes pour les mots ne sont paséquivalentes pour les permutations. Nousétudions la relation entre ces définitions en démontrant qu'ils constituent une chaîne d'inclusions. En outre, nous discutons les automates engendrant la permutation de Thue-Morse.

Enumeration and Decidable Properties of Automatic Sequences

“…For our final application, Frid and Zamboni [19] introduced the notion of "automatic permutation". This is a permutation of N based on a k-automatic sequence x, as follows: we say i < j if the infinite word x[i..∞] is lexicographically less than the word x[j..∞].…”

Minimal complexity of equidistributed infinite permutations

“…Other results on the comparison of words and permutations include discussions of automatic permutations [12] and of the Fine and Wilf theorem [11], and a study of square-free permutations [6].…”