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 formA 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.

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Abstract. We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-brokencircuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.Mathematics Subject Classifications (2010): Primary 05C22; Secondary 05A19, 05C05, 05C30, 52C35.