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A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and P-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a negative integer (and so, a priori the counting function doesarXiv:1201.2212v1 fatcat:tzbm2preknhmbcbjwoalt6itry