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arXiv · 2607.00674

Relaxation of Square-Freeness

Hiroki Shibata·Takuya Mieno·Dominik Köppl·Shunsuke Inenaga
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Abstract

We extend the analysis of nonrepetitive sequences of Entringer et al. [Journal of Combinatorial Theory, 1974] to relaxations of equality testing under nonstandard equivalence relations, in particular parameterized equivalence and order-preserving equivalence. For this setting, we introduce $\ell^+$-squares, defined as squares whose total length is at least $2\ell$. We obtain an infinite $3^+$-parameterized-square-free ternary word and an infinite $3^+$-order-preserving-square-free binary word through an approach based on combinatorics on words. In addition, we report the longest $\ell^+$-square-free words across several equivalence relations.

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Hiroki Shibata, Takuya Mieno, Dominik Köppl, Shunsuke Inenaga. 2026-07-01. Relaxation of Square-Freeness. https://arxiv.org/abs/2607.00674

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