April 11, 2005
This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.
Similar papers 1
October 26, 2009
This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer realizations. In particular, the paper presents two algorithms for computing the vector partition function with respect to a finite set of vectors $I$ as a quasipolynomial over a finite set of pointed polyhedral cones. We use the developed techniques to relate a result of P. Tumarkin an...
August 8, 2014
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of...
November 7, 2010
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A....
May 16, 2007
We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.
February 28, 2011
This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.
June 13, 2019
The Baldoni--Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function fundamental in representation theory. On the other hand the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an ...
May 29, 2014
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a couting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorit...
April 10, 2019
In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type A.
October 2, 2017
The Lidskii formula for the type $A_n$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope the volume is the leading coefficient of the Ehrhart polynomial. The beauty of the Lidskii formula is the revelation that for these polytopes its Ehrhart polynomial function can be deduced from its volume function! Baldoni and Vergne genera...
March 1, 2011
In this paper we study the classification problem of convex lattice ploytopes with respect to given volume or given cardinality.