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arXiv · 1308.2698

Positroids and non-crossing partitions

Federico Ardila·Felipe Rincón·Lauren Williams
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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. 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.

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Federico Ardila, Felipe Rincón, Lauren Williams. 2013-08-12. Positroids and non-crossing partitions. https://arxiv.org/abs/1308.2698

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