Ranking, unranking and random generation of extensional acyclic digraphs

Extensional acyclic digraphs are acyclic digraphs whose vertices have pairwise different sets of out‐neighbors; they represent hereditarily finite sets, which stand at the basis of some computer languages. In this paper we give an $O(n^3)$ algorithm for generating uniformly at random an extensional acyclic digraph on n vertices. This is done by first proposing a linear‐time algorithm for encoding such digraphs by particular $(n‐1)$ ‐tuples of subsets of $\{0,\dots ,n‐2\}$ . We then give a new counting recurrence for such tuples, which we exploit in ranking/unranking algorithms. These are also useful for indexing data structures by hereditarily finite sets.