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Positroids, non-crossing partitions, and positively oriented matroids
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 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 positive MacPhersonian) is homeomorphic to a closed ball.
Cited literature [23 references]
https://hal.inria.fr/hal-01207540
Contributor : Coordination Episciences Iam <>
Submitted on : Thursday, October 1, 2015 - 9:28:01 AM
Last modification on : Wednesday, July 22, 2020 - 9:14:03 AM
Long-term archiving on: : Saturday, January 2, 2016 - 10:34:00 AM
Contributor : Coordination Episciences Iam <>
Submitted on : Thursday, October 1, 2015 - 9:28:01 AM
Last modification on : Wednesday, July 22, 2020 - 9:14:03 AM
Long-term archiving on: : Saturday, January 2, 2016 - 10:34:00 AM
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