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Positroids and non-crossing partitions


Authors: Federico Ardila, Felipe Rincón and Lauren Williams
Journal: Trans. Amer. Math. Soc. 368 (2016), 337-363
MSC (2010): Primary 05A15, 05B35, 14M15, 14P10, 46L53
DOI: https://doi.org/10.1090/tran/6331
Published electronically: May 29, 2015
MathSciNet review: 3413866
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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 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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Additional Information

Federico Ardila
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132 – and – Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia
Email: federico@sfsu.edu

Felipe Rincón
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom – and – Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia
Email: e.f.rincon@warwick.ac.uk

Lauren Williams
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: williams@math.berkeley.edu

DOI: https://doi.org/10.1090/tran/6331
Received by editor(s): October 18, 2013
Received by editor(s) in revised form: November 5, 2013
Published electronically: May 29, 2015
Additional Notes: The first author was partially supported by the National Science Foundation CAREER Award DMS-0956178 and the SFSU-Colombia Combinatorics Initiative
The second author was supported by the EPSRC grant EP/I008071/1
The third author was partially supported by the National Science Foundation CAREER award DMS-1049513
Article copyright: © Copyright 2015 American Mathematical Society