2,420 research outputs found
Multidimensional Ehrhart Reciprocity
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized
Ehrhart's idea of counting lattice points in dilated rational polytopes: Given
a rational polytope, that is, a polytope with rational vertices, we use its
description as the intersection of halfspaces, which determine the facets of
the polytope. Instead of just a single dilation factor, we allow different
dilation factors for each of these facets. We proved that, if our polytope is a
simplex, the lattice point counts in the interior and closure of such a
vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type
reciprocity law. This generalizes the classical reciprocity law for rational
polytopes. In the present paper we complete the picture by extending this
result to general rational polytopes. As a corollary, we also generalize a
reciprocity theorem of Stanley.Comment: 7 page
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
An Extreme Family of Generalized Frobenius Numbers
We study a generalization of the \emph{Frobenius problem}: given positive
relatively prime integers, what is the largest integer that cannot be
represented as a nonnegative integral linear combination of these parameters?
More generally, what is the largest integer that has exactly such
representations? We illustrate a family of parameters, based on a recent paper
by Tripathi, whose generalized Frobenius numbers
exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \
g_{\binom{k+2}{k-1}}, ...g_{\binom{k+j}{k-1}}\binom{k+j+1}{k-}$
representations. Along the way, we introduce a variation of a generalized
Frobenius number and prove some basic results about it.Comment: 5 pages, to appear in Integers: the Electronic Journal of
Combinatorial Number Theor
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