Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form xj = xi + c are called affinographic. For an affinographic hyperplane arrangement in Rn, 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 vi has the form [hi + 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. Mathematics Subject Classifications (2000): Primary 05C22, 52C35; Secondary 05C15.