Paper Info
Title
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Abstract
Hyperplanes of the form x"j=x"i+c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(m) that counts integral points in [1,m]^n that do not lie in any hyperplane of the arrangement. We show that f(m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v"i has the form [h"i+1,m]. A related problem takes colors modulo m; the number of proper modular colorations is a different piecewise polynomial that for large m becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.
Year
DOI
Venue
2007
10.1016/j.jcta.2006.03.006
J. Comb. Theory, Ser. A
Keywords
Field
DocType
chromatic function,modular gain graph,affinographic hyperplane arrangement,piecewise polynomial function,deformation of coxeter arrangement,shi arrangement,positive integers m,colors modulo m,characteristic polynomial,proper coloring,linial arrangement,integral gain graph,interval graph coloring,integral point,large m,different piecewise polynomial,lattice point count,m increase,lattice points,interval graph
Integer,Discrete mathematics,Characteristic polynomial,Combinatorics,Gain graph,Vertex (geometry),Polynomial,Integral graph,Hyperplane,Mathematics,Piecewise
Journal
Volume
Issue
ISSN
114
1
Journal of Combinatorial Theory, Series A
Citations 
PageRank 
References 
3
0.50
8
Authors
2
Name
Order
Citations
PageRank
David Forge1369.49
T. Zaslavsky229756.67