On Periodicity and Complexity of Generalized Pseudostandard Words

TL;DR

This paper investigates the periodicity and complexity of generalized pseudostandard words, providing a precise condition for their periodicity and challenging a previous conjecture about their complexity bounds.

Contribution

It establishes a necessary and sufficient condition for the periodicity of generalized pseudostandard words and provides a counterexample to a prior complexity conjecture.

Findings

A new criterion for periodicity of generalized pseudostandard words.

Counterexample disproving the conjecture that complexity C(n) ≤ 4n for large n.

Enhanced understanding of the structure and complexity of these words.

Abstract

Generalized pseudostandard words have been introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. We present two new results concerning these words. The first one is a necessary and sufficient condition for their periodicity. The second result is a counterexample to Conjecture 43 from the paper: A. B. Masse, G.Paquin, H. Tremblay, and L. Vuillon, On Generalized Pseudostandard Words over Binary Alphabet (Journal of Int. Sequences, 16:Article 13.2.11, 2013) that estimated the complexity of binary generalized pseudostandard words as C(n) being less than or equal to 4n for all sufficiently large n.