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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The short toric polynomial


Author: Gábor Hetyei
Journal: Trans. Amer. Math. Soc. 365 (2013), 1441-1468
MSC (2010): Primary 06A07; Secondary 05A15, 06A11, 52B05
Published electronically: September 12, 2012
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Abstract: We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $ h$-vector in terms of the $ cd$-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric $ h$-vector of a dual simplicial Eulerian poset in terms of its $ f$-vector. This formula implies Gessel's formula for the toric $ h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $ h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.


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Additional Information

Gábor Hetyei
Affiliation: Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, North Carolina 28223
Email: ghetyei@uncc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05659-5
PII: S 0002-9947(2012)05659-5
Keywords: Eulerian poset, toric $h$-vector, Narayana numbers, reflection principle, Morgan-Voyce polynomial
Received by editor(s): November 16, 2010
Received by editor(s) in revised form: June 22, 2011
Published electronically: September 12, 2012
Article copyright: © Copyright 2012 American Mathematical Society