138 research outputs found
Stanley's Major Contributions to Ehrhart Theory
This expository paper features a few highlights of Richard Stanley's
extensive work in Ehrhart theory, the study of integer-point enumeration in
rational polyhedra. We include results from the recent literature building on
Stanley's work, as well as several open problems.Comment: 9 pages; to appear in the 70th-birthday volume honoring Richard
Stanle
A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope
We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric
The boundary volume of a lattice polytope
For a d-dimensional convex lattice polytope P, a formula for the boundary
volume is derived in terms of the number of boundary lattice points on the
first \floor{d/2} dilations of P. As an application we give a necessary and
sufficient condition for a polytope to be reflexive, and derive formulae for
the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give
applications to reflexive order polytopes, and to the Birkhoff polytope.Comment: 21 pages; subsumes arXiv:1002.1908 [math.CO]; to appear in the
Bulletin of the Australian Mathematical Societ
On positivity of Ehrhart polynomials
Ehrhart discovered that the function that counts the number of lattice points
in dilations of an integral polytope is a polynomial. We call the coefficients
of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive
if all Ehrhart coefficients are positive (which is not true for all integral
polytopes). The main purpose of this article is to survey interesting families
of polytopes that are known to be Ehrhart positive and discuss the reasons from
which their Ehrhart positivity follows. We also include examples of polytopes
that have negative Ehrhart coefficients and polytopes that are conjectured to
be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic
Combinatorics, a volume of the Association for Women in Mathematics Series,
Springer International Publishin
Magic graphs and the faces of the Birkhoff polytope
Magic labelings of graphs are studied in great detail by Stanley and Stewart.
In this article, we construct and enumerate magic labelings of graphs using
Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes.
We define polytopes of magic labelings of graphs and digraphs. We give a
description of the faces of the Birkhoff polytope as polytopes of magic
labelings of digraphs.Comment: 9 page
Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices
In Ehrhart theory, the -vector of a rational polytope often provide
insights into properties of the polytope that may be otherwise obscured. As an
example, the Birkhoff polytope, also known as the polytope of real
doubly-stochastic matrices, has a unimodal -vector, but when even small
modifications are made to the polytope, the same property can be very difficult
to prove. In this paper, we examine the -vectors of a class of polytopes
containing real doubly-stochastic symmetric matrices.Comment: 11 pages; this revision removes an erroneous proposition from earlier
versions and expands on the implication
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