Inside-out polytopes

Abstract

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.

An Enumerative Geometry for Magic and Magilatin Labellings

“…This is a quasipolynomial in t by the general theory of inside-out polytopes. Then inside-out reciprocity [4] gives the enumeration of weak nonnegative squares with multiplicity; this is reminiscent of Stanley's theorem on acylic orientations [14].…”

“…With the theory of inside-out polytopes [4] we can attack this and related counting problems in a systematic way obtaining a general result about magic counting functions and an interpretation of reciprocity that leads to a new kind of question about permutations. In inside-out theory we supplement the polytope P = [0, 1] n 2 ∩ s with the pair-equality hyperplane arrangement…”

“…The theory of inside-out polytopes [4] is designed to count those points of the integral lattice Z d that are contained in a rational convex polytope P but not in a rational affine hyperplane arrangement H, that is, where each hyperplane is spanned by the rational points it contains. We call (P, H) a rational inside-out polytope, closed if P is closed.…”

Bounds on the coefficients of tension and flow polynomials

“…These constructions are from [2,3,12]. The inside-out polytopes ker A ∩ (−1, 1) E , H and ker M ∩ (−1, 1) E , H are integral (which follows from the total unimodularity of A and M, see [22]) and compressed (which follows e.g.…”

“…To this end we employ Ehrhart theory, the theory of lattice points in polyhedra. In past research [2,3,5,7,12] it has been shown that each of these polynomials can be realized as the Ehrhart polynomial of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex C ⊆ C. In this article we show that for a wide class of inside-out polytopes, this polytopal complex can be triangulated so that the resulting relative simplicial complex ⊆ is unimodular and admits a convex ear decomposition (Theorem 2).…”