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Séminaire Lotharingien de Combinatoire, 80B.44 (2018), 12 pp.

# Felix Gotti

# On 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 **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*). Positroids can be parameterized by several families of
combinatorial objects. Here we characterize some of these families for
the positroids we produce, namely, decorated permutations,
Le-diagrams, and move-equivalence classes of plabic
graphs. Finally, we describe the matroid polytope of a given rational
Dyck positroid.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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