Asymptotic estimates for double-coverings

A collection of finite sets $\{A_1,A_2,\dots ,A_p\}$ is said to be a double-covering if each $a\in {\cup }_{k=1}^p{A}_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let ${\mu }_{l,p}$ be the number of equivalency classes of double-coverings with $\mathrm{\#}(A_k)=l$, $k=1,2,\dots ,p$. We characterize the asymptotic behavior of the quantity ${\mu }_{l,p}$ as $p\to \mathrm{\infty }$. The results are applied to give an alternative approach to the Bonami-Kiener hypercontraction inequality.