On the number of n-ary quasigroups of finite order

Abstract

Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a recurrent formula for Q(n,4). We prove that for all $n\ge 2$ and $k\ge 5$ the following inequalities hold: $(k-3/2{)}^{n/2}(\frac{k-1}{2}{)}^{n/2}<lo{g}_{2}Q(n,k)\le c_k(k-2{)}^n$, where $c_k$ does not depend on $n$. So, the upper asymptotic bound for $Q(n,k)$ is improved for any $k\ge 5$ and the lower bound is improved for odd $k\ge 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.

english 9pp, russian 9pp. v.2: corrected: initial data for recursion; added: Appendix with program

References in corpus (1)

• On reconstructing reducible n-ary quasigroups and switching subquasigroups

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