102 research outputs found
Ehrhart -vectors of hypersimplices
We consider the Ehrhart -vector for the hypersimplex. It is well-known
that the sum of the is the normalized volume which equals an Eulerian
numbers. The main result is a proof of a conjecture by R. Stanley which gives
an interpretation of the coefficients in terms of descents and
excedances. Our proof is geometric using a careful book-keeping of a shelling
of a unimodular triangulation. We generalize this result to other closely
related polytopes
Morse Matchings on a Hypersimplex
We present a family of complete acyclic Morse matchings on the face lattice
of a hypersimplex. Since a hypersimplex is a convex polytope, there is a
natural way to form a CW complex from its faces. In a future paper we will
utilize these matchings to classify every subcomplex whose reduced homology
groups are concentrated in a single degree and describe a homology basis for
each of them.Comment: part of phD thesis arXiv:1108.6001. arXiv admin note: text overlap
with arXiv:0806.1503 by other author
Positive configuration space
We define and study the totally nonnegative part of the Chow quotient of the
Grassmannian, or more simply the nonnegative configuration space. This space
has a natural stratification by positive Chow cells, and we show that
nonnegative configuration space is homeomorphic to a polytope as a stratified
space. We establish bijections between positive Chow cells and the following
sets: (a) regular subdivisions of the hypersimplex into positroid polytopes,
(b) the set of cones in the positive tropical Grassmannian, and (c) the set of
cones in the positive Dressian. Our work is motivated by connections to super
Yang-Mills scattering amplitudes, which will be discussed in a sequel.Comment: 46 pages; citations adde
The -Construction for Lattices, Spheres and Polytopes
We describe and analyze a new construction that produces new Eulerian
lattices from old ones. It specializes to a construction that produces new
strongly regular cellular spheres (whose face lattices are Eulerian). The
construction does not always specialize to convex polytopes; however, in a
number of cases where we can realize it, it produces interesting classes of
polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple
4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct
for each an infinite family of -simplicial 2-simple
-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
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