Infinite Lyndon words

In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b are odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/( √ 2b − 1 + 1).

“…Theorem 2 ( [15]). Any infinite word w is uniquely expressed as a non-increasing product of Lyndon words, finite or infinite, in one of the two following forms:…”

Section: Preliminariesmentioning

confidence: 97%

“…Siromoney et al [15] extended Theorem 1 to infinite words. The set L ∞ of infinite Lyndon words consists of words smaller than any of their suffixes.…”

Section: Preliminariesmentioning

confidence: 97%

Smooth words on 2-letter alphabets having same parity

Brlek¹,

Jamet²,

Paquin³

2008

“…Siromoney et al showed in [13] that the infinite Lyndon words are precisely the limits of prefix-preserving increasing sequences of finite Lyndon words. We show that this result still holds when Lyndon words are replaced with generalized Lyndon words provided that the infinite word is primitive.…”

Section: Characterization Of Infinite Generalized Lyndon Wordsmentioning

confidence: 99%

“…About 40 years later, infinite Lyndon words were introduced in [13]. There are several equivalent definitions, but we use the definition which focuses on the idea of rotation.…”

Section: Introductionmentioning

confidence: 99%

Generalized Lyndon factorizations of infinite words

Burcroff

Winsor

2020

A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We prove a conjecture of Dolce, Restivo, and Reutenauer: every infinite word has a unique nonincreasing factorization into finite and infinite generalized Lyndon words. When this factorization has finitely many terms, we characterize the last term of the factorization. Our methods also show that the infinite generalized Lyndon words are precisely the words with infinitely many generalized Lyndon prefixes.Keywords: generalized lexicographic order · infinite generalized Lyndon word · unique nonincreasing Lyndon factorization arXiv:1905.04746v2 [math.CO]

“….. Thus, following [3] and [19], we say that an infinite word x is a generalized infinite Lyndon word if x is smaller that any of its nontrivial proper suffixes. It would be interesting to generalize this result to Galois words and other generalized Lyndon words.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%

On generalized Lyndon words

Dolce

Restivo

Reutenauer

2019

A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincreasing factorization of generalized Lyndon words. We give new characterizations of the first and the last factor in this factorization as well as new characterization of generalized Lyndon words. We also give more specific results on two special cases: the classical one and the one arising from the alternating lexicographical order.