Abstract
Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.
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References
Beck M., Robins S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer-Verlag, New York (2007)
Braun, B., Solus, L.: Shellability, Ehrhart theory, and r-stable hypersimplices. Preprint. available at: http://arxiv.org/pdf/1408.4713v3.pdf (2016)
Bruns W., Römer T.: h-vectors of Gorenstein polytopes. J. Combin. Theory Ser. A 114(1), 65–76 (2007)
De Loera J., Haws D., Köppe M.: Ehrhart polynomials of matroid polytopes and polymatroids. Discrete Comput. Geom. 42(4), 670–702 (2009)
De Negri E., Hibi T.: Gorenstein algebras of Veronese type. J. Algebra 193(2), 629–639 (1997)
Hibi T.: Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237–240 (1992)
Katzman M.: The Hilbert series of algebras of the Veronese type. Comm. Algebra 33(4), 1141–1146 (2005)
Lam T., Postnikov A.: Alcoved polytopes, I. Discrete Comput. Geom. 38(3), 453–478 (2007)
Stanley R.: A monotonicity property of h-vectors and h*-vectors. European J. Combin. 14(3), 251–258 (1993)
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Liam Solus was supported by a 2014 National Science Foundation/Japan Society for the Promotion of Science East Asia and Pacific Summer Institute Fellowship.
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Hibi, T., Solus, L. Facets of the r-Stable (n, k)-Hypersimplex. Ann. Comb. 20, 815–829 (2016). https://doi.org/10.1007/s00026-016-0325-x
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DOI: https://doi.org/10.1007/s00026-016-0325-x