On the Enumeration and Asymptotic Growth of Free Quasigroup Words

TL;DR

This paper introduces peri-Catalan numbers to count reduced quasigroup words, provides a recursive formula based on the Euclidean Algorithm, and explores their asymptotic growth and irrelevance of identities.

Contribution

It presents the first recursive formula for peri-Catalan numbers and investigates their asymptotic behavior, highlighting the negligible effect of quasigroup identities.

Findings

Recursive formula for peri-Catalan numbers derived from Euclidean Algorithm

Numerical evidence supporting conjectured asymptotic growth

Asymptotic irrelevance of quasigroup identities for long words

Abstract

The 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 the 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.