2,599 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
A Meshalkin theorem for projective geometries
Let M be a family of sequences (a_1,...,a_p) where each a_k is a flat in a
projective geometry of rank n (dimension n-1) and order q, and the sum of
ranks, r(a_1) + ... + r(a_p), equals the rank of the join a_1 v ... v a_p. We
prove upper bounds on |M| and corresponding LYM inequalities assuming that (i)
all joins are the whole geometry and for each k<p the set of all a_k's of
sequences in M contains no chain of length l, and that (ii) the joins are
arbitrary and the chain condition holds for all k. These results are q-analogs
of generalizations of Meshalkin's and Erdos's generalizations of Sperner's
theorem and their LYM companions, and they generalize Rota and Harper's
q-analog of Erdos's generalization.Comment: 8 pages, added journal referenc
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