Asymptotic estimates for double-coverings

A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ 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 $\#(A_k)=l$, $k=1,2,\ldots,p$. We characterize the asymptotic behavior of the quantity $\mu_{l,p}$ as $p\to \infty$. The results are applied to give an alternative approach to the Bonami-Kiener hypercontraction inequality.