AI 中文总结
本文扩展了非重复序列分析，引入参数化等价和保序等价下的松弛平方自由性，通过形态构造得到无限长的3^+参数化平方自由三元词和3^+保序平方自由二元词。
AI 中文摘要
我们将Entringer等人[Journal of Combinatorial Theory, 1974]对非重复序列的分析扩展到非标准等价关系下相等性检验的松弛，特别是参数化等价和保序等价。针对这一设定，我们引入$\ell^+$-平方，定义为总长度至少为$2\ell$的平方。通过形态构造，我们得到一个无限长的$3^+$-参数化平方自由三元词和一个无限长的$3^+$-保序平方自由二元词。此外，我们报告了多种等价关系下最长的$\ell^+$-平方自由词。
英文摘要
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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