non-uniform hypergraph $G$ on $n$ vertices, we define the Lubell function of $G$ as $$\label{eq:lubell}
    h_{n}(G):=\sum_{F\in E(G)}\frac{1}{\binom{n}{|F|}}=\sum_{k\in R(G)}\frac{|E(H^{k})|}{\binom{n}{k}}.$$

The Lubell function is the expected number of edges hit by a random full chain. Namely, pick a uniformly random permutation $\sigma$ on $n$ vertices; define a random full chain $C_\sigma$ by $$\{ \{\emptyset\}, \{\sigma(1)\}, \{\sigma(1), \sigma(2)\}, \cdots, [n]\}.$$ Let $X=|E(G)\cap C_{\sigma}|$, the number of edges hit by the random full chain. Then $$\label{eq:X}
  h_n(G)={{\rm E}}(X).$$

Given two hypergraphs $H_1$ and $H_2$, we say $H_1$ is a subgraph of $H_2$, denoted by $H_1\subseteq H_2$, if there exists a 1-1 map $f\colon V(H_1)\to V(H_2)$ so that $f(F)\in E(H_2)$ for any $F\in E(H_1)$. Whenever this occurs, we say the image $f(H_1)$ is an [*ordered copy*]{} of $H_2$, written as $H_1\stackrel{f}{\hookrightarrow} H_2$. A necessary condition for $H_1\subseteq H_2$ is $R(H_1)\subseteq R(H_2)$. Given a subset $K\subseteq V(H)$ and a subset $S\subseteq R(H)$, the [*induced subgraph*]{}, denoted by $H^{[S]}[K]$, is a hypergraph on $K$ with the edge set $\{F\in E(H)\colon F\subseteq K \mbox{ and } |F|\in S\}$. When $S=R(H)$, we simply write $H[K]$ for $H^{[S]}[K]$.

Given a positive integer $n$ and a subset $R\subseteq [n]$, the complete hypergraph $K^R_{n}$ is a hypergraph on $n$ vertices with edge 