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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.
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Federico Ardila, Felipe Rincón, Lauren Williams. Positroids, non-crossing partitions, and positively oriented matroids. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.655-666. ⟨hal-01207540⟩

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