Positroids and non-crossing partitions

Abstract

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [ n ] [n] is connected equals 1 / e 2 1/e^2 asymptotically.