Let $V$ be a set consisting of $n$ points, $n\geq 2$.

A Sperner family on $V$ is a set $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$, $m\geq 1$, where each $\sigma_i$, $i=1,2,\dots,m $, is a nonempty subset of $V$, and for any $i\neq j$, $\sigma_i$ does not contain $\sigma_j$.

Given a Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$, an extension of it is a new Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m,\sigma_{m+1}\}$ obtained by adding a subset $\sigma_{m+1}$ of $V$ to $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$.

**Question:** Let $R(\sigma_1,\sigma_2,\dots,\sigma_m)$ be the number of extensions of the Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$. Are there any methods or formulas to compute $R(\sigma_1,\sigma_2,\dots,\sigma_m)$? Are there any references?

Thanks.