Positroids Induced by Rational Dyck Paths

Abstract: A rational Dyck path of type $(m,d)$ is an increasing unit-step lattice path from $(0,0)$ to $(m,d) \in \mathbb{Z}^2$ that never goes above the diagonal line $y = (d/m)x$. On the other hand, a positroid of rank $d$ on the ground set $[d+m]$ is a special type of matroid coming from the totally nonnegative Grassmannian. In this paper we describe how to naturally assign a rank $d$ positroid on the ground set $[d+m]$, which we name rational Dyck positroid, to each rational Dyck path of type $(m,d)$. We show that such an assignment is one-to-one. There are several families of combinatorial objects in one-to-one correspondence with the set of positroids. Here we characterize some of these families for the positroids we produce, namely Grassmann necklaces, decorated permutations, Le-diagrams, and move-equivalence classes of plabic graphs. Finally, we describe the matroid polytope of a given rational Dyck positroid.