Unranking paths in a graph/lattice
Suppose without loss of generality that there is one source $\emptyset$ and one sink $S$. For each node $v$ in your digraph, you can compute recursively the number $N_v$ of unique maximum length paths from $\emptyset$ to $v$: at $\emptyset$ we have $N_\emptyset = 1$, and at other points, $N_v = \sum_u N_u$, where the sum goes over all immediate predecessors of $u$.

This suggests the following encoding for maximal paths at a node $v$. Suppose the immediate predecessors of $v$ are $u_1,\ldots,u_k$, in some arbitrary but fixed order. We will encode all paths as numbers in the range $[0,N_v)$ ($=\{0,\ldots,N_v-1\}$) in the following way: all paths through $u_1$ will be encoded in $[0,N_{u_1})$, all paths through $u_2$ will be encoded in $[N_{u_1},N_{u_1}+N_{u_2})$, and so on. Inside each interval, we apply the same encoding scheme recursively. By applying this scheme for $v = S$ (the sink), we get an efficient encoding system for all maximal paths.