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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Positroids and non-crossing partitions
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by Federico Ardila, Felipe Rincón and Lauren Williams PDF
Trans. Amer. Math. Soc. 368 (2016), 337-363 Request permission

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
  • MR Author ID: 725066
  • 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
  • MR Author ID: 611667
  • Email: williams@math.berkeley.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3413866