Positroids and non-crossing partitions

By Federico Ardila, Felipe Rinc'on, Lauren K. Williams. 2013

78 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Materials Chemistry, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/85e778cf596683faa5960e1fa9e5be86dd1acb33

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In Discrete Mathematics and Combinatorics

In Materials Chemistry

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Citations in the corpus

The positive Dressian equals the positive tropical Grassmannian (2020, background): “nor which is the uniform matroid U 2;4 or the graphical matroid M K 4 associated to the complete graph K 4. The graphical matroid M K 4 is not a positroid, and all minors of positroids are positroids [ARW16], so we have the following corollary. Corollary 6.4. A connected positroid is series-parallel”
Bruhat Interval Polytopes (2014, methodology): “ a polytope which is obtained by moving the vertices of the usual permutohedron in such a way that directions of edges are preserved, but some edges (and higher dimensional faces) may degenerate. See [ARW13] and [Pos09] for more details on positroid polytopes and generalized permutohedra. There is a”