Unranking paths in a graph/lattice

Question

A ranking algorithm determines the position (or rank) of a combinatorial object among all the objects (with respect to a given order); an unranking algorithm finds the object having a specified rank. Thus, ranking and unranking can be considered as inverse operations.

The following lattice has 9 unique max length paths from {} to {1,2,3,4,5}, which can be obtained by a depth first search.

(The graph is directed, with arrows pointing down)

Is it possible to write a function that generates the N'th path without enumerating paths 1..N-1.

see: https://math.stackexchange.com/questions/510911/computing-all-simple-paths-in-a-distributive-lattice-in-parallel for more problem details.