# Positroids, non-crossing partitions, 1/e^2, and a conjecture of Da Silva

# Positroids, non-crossing partitions, 1/e^2, and a conjecture of Da Silva

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. We use this to prove that the probability that a positroid on [n] is connected equals 1/e^2 asymptotically. We also prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. This is joint work with Felipe Rincón (Warwick) and Lauren Williams (Berkeley).