Transversals, plexes, and multiplexes in iterated quasigroups

A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$. A transversal is a 1-plex. In this paper we prove that there exists a constant $c(G,k)$ such that if a $d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$ the number of its $k$-multiplexes is asymptotically equal to $c(G,k) \left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of order $n$ is nonzero then asymptotically it is $c(G,1) n!^{d-1}$. In addition, we provide limit constants and recurrence formulas for the numbers of transversals in two iterated quasigroups of order 5, characterize a typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and transversals in $d$-iterated quasigroups.