(470,200)[1]{} (300,100)[(1,1)[300]{}]{} (320,250)[3]{} (0,400)[(0,1)[300]{}]{} (-80,500)[4]{} (0,400)[(1,1)[300]{}]{} (100,550)[2]{} (0,400)[(2,1)[600]{}]{} (120,400)[2]{} (300,400)[(0,1)[300]{}]{} (320,450)[1]{} (600,400)[(0,1)[300]{}]{} (620,500)[2]{} (600,400)[(-1,1)[300]{}]{} (450,465)[1]{} (0,700)[(1,1)[300]{}]{} (50,830)[1]{} (300,700)[(0,1)[300]{}]{} (320,800)[2]{} (600,700)[(-1,1)[300]{}]{} (530,830)[2]{} (300,1000)[(0,1)[300]{}]{} (320,1100)[6]{}

\[lem-rank\] There is a rank function on $\rho $ on $\mathcal{ELP}(G)$ such that $\rho(\Lambda_i) = \rho(\Lambda_j) +1$ whenever $\Lambda_i $ covers $\Lambda_j$.

Each $\Lambda_i$ in $\Lambda(G)$ is nonempty. Therefore, for each $\Lambda_i$ in $\Lambda(G)$ and for each $k$ such that $2 \leq k \leq v(\Lambda_i)$ there is at least one nonempty induced subgraph $\Lambda_j$ of $\Lambda_i$ such that $v(\Lambda_j) = k$. Moreover, empty induced subgraphs do not belong to $\Lambda(G)$. Therefore, $\rho(\Lambda_i) = v(\Lambda_i)$ meets the requirements of a rank function.

Stanley [@stanley1997] defines a rank function such that the $\rho(x) = 0$ for a minimal element $x$. But we have deviated from that convention since $\rho(\Lambda_i) = v(\Lambda_i)$ for each $\Lambda_i \in \Lambda(G)$ is more convenient here. We now demonstrate that $\mathcal{N}(G)$ and $\mathcal{ELP}(G)$ are really equivalent, that is, they can be constructed from each other.

\[lem-kelly1\] Let $F$ and $H$ be two graphs, and let $q$ be an integer such that $v(F) \leq q \leq v(H)$. Then $$\label{eq-kelly1}
\sum_{X| v(X) = q}\displaystyle\binom{H}{X}\displaystyle\binom{X}{F}
= \displaystyle\binom{v(H)-v(F)}{q-v(F)}\displaystyle\binom{H}{F}$$ where the summation is over distinct isomorphism types $X$.

This is 