2017/06/29 by Gotti, Felix
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.09921
A rational Dyck path of type (m,d) is an increasing unit-step lattice path from (0,0) to (m,d) ∈ ℤ2 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.