A problem equivalent to counting directed acyclic graphs on labeled vertices

Abstract

An encoding of directed acyclic graphs (DAGs) on labeled vertices is proposed, which is a generalisation of the Pr\"ufer code for labeled trees, if a certain orienation on the edges of the tree is introduced. Hence it is shown that the number of sequences $S_1, S_2, \ldots, S_{n-1}$ of subsets of $\{1, 2, \ldots, n\}$ with the property that $|\bigcup_{i=1}^kS_i|\leq k$ for every $1\leq k \leq n-1$, is equal to the number of DAGs on $n$ labeled vertices.