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Two unfortunate properties of pure $ f$-vectors


Authors: Adrián Pastine and Fabrizio Zanello
Journal: Proc. Amer. Math. Soc. 143 (2015), 955-964
MSC (2010): Primary 05E40; Secondary 13F55, 05E45, 05B07, 13H10
DOI: https://doi.org/10.1090/S0002-9939-2014-12338-0
Published electronically: October 16, 2014
MathSciNet review: 3293713
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Abstract | References | Similar Articles | Additional Information

Abstract: The set of $ f$-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure is very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact.

We first show that pure (in fact, Cohen-Macaulay) $ f$-vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure $ O$-sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure $ O$-sequences, we show that the interval property fails for the set of pure $ f$-vectors, even in dimension 2.


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Additional Information

Adrián Pastine
Affiliation: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931-1295
Email: agpastin@mtu.edu

Fabrizio Zanello
Affiliation: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931-1295
Email: zanello@math.mit.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12338-0
Keywords: Pure simplicial complex, Cohen-Macaulay complex, unimodality, pure $f$-vector, interval property, pure $O$-sequence, Steiner system
Received by editor(s): November 15, 2012
Received by editor(s) in revised form: June 10, 2013
Published electronically: October 16, 2014
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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