On the number of n-ary quasigroups of finite order

Details On the number of n-ary quasigroups of finite order On the number of n-ary quasigroups of finite order

2009-12-30

arxiv.org/abs/0912.5453v1

Mathematics

Combinatorics

english 8pp, russian 8pp

Scientific paper

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 $n\geq 2$ and $k\geq 5$ the following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k) \leq {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 $k\geq 5$ and the lower bound is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.

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