A new estimate on complexity of binary generalized pseudostandard words

TL;DR

This paper challenges previous complexity bounds for binary generalized pseudostandard words by providing a counterexample to an existing conjecture and proposing a new conjecture with a higher upper bound.

Contribution

It presents a counterexample to a prior complexity conjecture and introduces a new conjecture suggesting a different upper bound for these words.

Findings

Counterexample disproves the ${ m C}(n) \,\leq\, 4n$ bound

Proposes a new conjecture ${ m C}(n) < 6n$ for all natural numbers

Advances understanding of the complexity of generalized pseudostandard words

Abstract

Generalized pseudostandard words were 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 a counterexample to Conjecture 43 from a paper by Blondin Mass\'e et al. that estimated the complexity of binary generalized pseudostandard words as ${\mathcal C}(n) \leq 4n$ for all sufficiently large $n$. We conjecture that ${\mathcal C}(n)<6n$ for all $n \in \mathbb N$.