  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.
