  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.
