On the number of invertible homogeneous structures

We estimate the number $r(n,m)$ of functions of $n$-valued logic in $m+1$ variables which are local transition functions of reversible homogeneous structures with arbitrary fixed neighbourhood pattern consisting of $m$ vectors. It follows from the results obtained in the paper that if $n\to\infty$, then
$$ \ln r(n,m)\sim n^{m+1}\ln n $$
uniformly in $m$.
This research was supported by the Russian Foundation for Basic Research, grant 02–01–00162.