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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Dehn–Sommerville relations and the Catalan matroid
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by Anastasia Chavez and Nicole Yamzon PDF
Proc. Amer. Math. Soc. 145 (2017), 4041-4047 Request permission

Abstract:

The $f$-vector of a $d$-dimensional polytope $P$ stores the number of faces of each dimension. When $P$ is a simplicial polytope the Dehn–Sommerville relations condense the $f$-vector into the $g$-vector, which has length $\lceil {\frac {d+1}{2}}\rceil$. Thus, to determine the $f$-vector of $P$, we only need to know approximately half of its entries. This raises the question: Which $(\lceil {\frac {d+1}{2}}\rceil )$-subsets of the $f$-vector of a general simplicial polytope are sufficient to determine the whole $f$-vector? We prove that the answer is given by the bases of the Catalan matroid.
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Additional Information
  • Anastasia Chavez
  • Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall 3840, Berkeley, California 94720
  • Email: a.chavez@berkeley.edu
  • Nicole Yamzon
  • Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132
  • Email: nyamzon@mail.sfsu.edu
  • Received by editor(s): December 26, 2015
  • Received by editor(s) in revised form: August 18, 2016, and October 4, 2016
  • Published electronically: March 23, 2017
  • Additional Notes: The first author was supported in part by NSF Alliances for Graduate Education and the Professoriate
    The second author was supported in part by NSF GK-12 grant DGE-0841164
  • Communicated by: Patricia Hersh
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4041-4047
  • MSC (2010): Primary 52B05, 52B40
  • DOI: https://doi.org/10.1090/proc/13554
  • MathSciNet review: 3665055