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arXiv · 2210.00508

The lexicographically least square-free word with a given prefix

Siddharth Berera·Andrés Gómez-Colunga·Joey Lakerdas-Gayle·John López·Mauditra Matin·Daniel Roebuck·Eric Rowland·Noam Scully·Juliet Whidden
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Abstract

The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $n\geq3$, and $L(p)$ for most families of two-letter words $p$.

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Siddharth Berera, Andrés Gómez-Colunga, Joey Lakerdas-Gayle, John López, Mauditra Matin, Daniel Roebuck, Eric Rowland, Noam Scully, Juliet Whidden. 2022-10-02. The lexicographically least square-free word with a given prefix. https://doi.org/10.37236/11659

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