This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

中文翻译：

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自由拟群词的枚举与渐近增长

本文统计了一定数量的生成器中特定长度的约简拟群词的数量。考虑到与加泰罗尼亚数的关系，计算自由岩浆中的单词，我们引入术语 peri-Catalan 数来表示自由准群字数。本文的主要成果是一个精确的 peri-Catalan 数递归公式，由欧几里得算法构造。欧几里得算法结构不容易适用于渐近分析的标准技术。然而，提出了由数值数据证实的 peri-Catalan 数的渐近行为的猜想。观察到的渐近行为的一个显着方面是所谓的准群恒等式渐近无关，