  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 freely 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] is
connected equals 1/e^2 asymptotically.

Combinatorics