Federico Ardila ; Felipe Rincón ; Lauren Williams

Positroids, noncrossing partitions, and positively oriented matroids
dmtcs:2431 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2014,
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

https://doi.org/10.46298/dmtcs.2431
Positroids, noncrossing partitions, and positively oriented matroids
Authors: Federico Ardila ^{1,}^{2}; Felipe Rincón ^{3}; Lauren Williams ^{4}
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Federico Ardila;Felipe Rincón;Lauren Williams
1 Department of Mathematics [San Francisco]
2 Universidad de los Andes [Bogota]
3 Warwick Mathematics Institute
4 Department of Mathematics [Berkeley]
We investigate the role that noncrossing 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 noncrossing 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 enumerate connected positroids, and we 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 <i>positive MacPhersonian</i>) is homeomorphic to a closed ball.