Characteristic and Ehrhart Polynomials

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.

We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so, we develop a generalization of Dumont’s surjective staircases, and introduce a polynomial which enumerates these generalized surjective staircases according to several statistics. We prove a recurrence for these polynomials and show that in certain special cases this recurrence can be solved explicitly to yield a generating function. We also prove refined versions of several of these results using the theory of complex hyperplane arrangements.

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.

We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

A note on isotropic discrepancy and spectral test of lattice point sets

Separable elements in Weyl groups

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

On the algebra associated with a geometric lattice

Topology quasi-preservation by local parallel operations

We give a representation-theoretic (or rather, crystal-theoretic) proof of Cherednik-Orr's recursion formula of Demazure type for the specialization at t = ∞ of the nonsymmetric Macdonald polynomials Ewλ(q, t), w ∈ W, where λ is a dominant integral weight and W is a finite Weyl group.

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.

Topological methods are described for estimating the size and depth of decision trees where a linear test is performed at each node. The methods are applied, among others, to the questions of deciding by a linear decision tree whether given n n real numbers (1) some k k of them are equal, or (2) some k k of them are unequal. We show that the minimum depth of a linear decision tree for these problems is at least (1) max { n − 1 , n lo g 3 ( n / 3 k ) } {\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(n/3k)\} , and (2) max { n − 1 , n lo g 3 ( k − 1 ) − k + 1 } {\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(k - 1) - k + 1\} . Our main lower bound for the size of linear decision trees for polyhedra P P in R n {{\mathbf {R}}^n} is given by the sum of Betti numbers for the complement R n ∖ P {{\mathbf {R}}^n}\backslash P . The applications of this general topological bound involve the computation of the Möbius function of intersection lattices of certain subspace arrangements. In particular, this leads to computing various expressions for the Möbius function of posets of partitions with restricted block sizes. Some of these formulas have topological meaning. For instance, we derive a formula for the Euler characteristic of the subset of R n {{\mathbf {R}}^n} of points with no k k coordinates equal in terms of the roots of the truncated exponential ∑ i > k x i / i ! \sum \nolimits _{i > k} {{x^i}} /i! .

Good lattice point sets are an important kind of low discrepancy points for multidimensional quadrature, simulation, experimental design, etc. The theoretical development of lattice point sets began over 40 years ago, but some important gaps in the theory remain. This article reviews the development of lattice point sets and highlights some open problems.

An improved algorithm for checking the injectivity of 2D toric surface patches

Let $V = \sum\nolimits_{i = 1}^n {Z / (k_i + 1)},0\leqq k_1 \leqq k_2 \leqq \cdots \leqq k_n \leqq \infty $, be the set of discrete lattice points in a torus. Let $e_i $ be the unit vector along the ith axis, $i = 1,2, \cdots ,n,$ and let ${\operatorname{Bdry}}(A) = \{ x \in V| {x \notin A,x - e_i \in A\,\,{\text{for some}}\,i} \} $. In this paper we obtain extremal configurations that minimize the number of boundary points among sets of given size in V. We show further that the result obtained implies the generalized Macaulay theorem of Clements and Lindström and the well-known Kruskal–Katona theorem.