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.