Eulerian digraphs and Dyck words, a bijection

The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian digraphs exploits a novel combinatorial structure: a binary matrix, we call Dyck matrix, representing the cycles of an Eulerian digraph. 1. Background, and motivation A digraph G is Eulerian if at every vertex the in-degree equals the out-degree. (Note that we do not require G to be connected.) The edge set of an Eulerian digraph G can be partitioned into directed cycles. For a non-empty multiset s = {s1, s2, . . . , sn}, of n positive integers, we call an Eulerian digraph s-labelled if its edge set is partitioned into n directed cycles of length s1, s2, . . . , sn, each with a distinguished first edge (and hence a unique second, third, etc., m-th edge). Figure 1 shows a {3, 2, 1}-labelled Eulerian digraph, with its 3 directed cycles of size 1, 2 and 3; the j edge of the i cycle is labelled ei,j . (Notice that in next Sections we endow these digraphs with a linear order on their cycles.)