Extremal generalized smooth words

Abstract

International audienceIn this article, we consider smooth words over 2-letter alphabets {a, b}, with a, b 2 N having same parity. We show that they all are recurrent and provide a linear algorithm computing the extremal words. Moreover, the set of factors of any infinite smooth word over an odd alphabet is closed under reversal, while it is not for even parity alphabets. The minimal word is an infinite Lyndon word if and only if either a = 1 and b odd, or a, b even. We also describe a connection between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets. Finally, the density of letters in extremal words is 1/2 for even alphabets, and 1/(p2b − 1 − 1) for a = 1 with b odd