A common theme of enumerative combinatorics is formed by counting functionsthat are polynomials evaluated at positive integers. In this expository paper,we focus on four families of such counting functions connected to hyperplanearrangements, lattice points in polyhedra, proper colorings of graphs, and$P$-partitions. We will see that in each instance we get interestinginformation out of a counting function when we evaluate it at a emph{negative}integer (and so, a priori the counting function does not make sense at thisnumber). Our goals are to convey some of the charm these "alternative"evaluations of counting functions exhibit, and to weave a unifying threadthrough various combinatorial reciprocity theorems by looking at them throughthe lens of geometry, which will include some scenic detours through othercombinatorial concepts.
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机译：通过计数函数函数形成突出组合物的共同主题是在正整数处评估的多项式。在本展品中，我们专注于与超平坦地区的这种计数函数的四个家庭，Polyhedra的格子点，图表的适当着色和$ P $ -Partition。我们将在每个实例中看到，当我们在A Emph {否定}整数（且，先验计数函数未在ThisNumber中没有意义时，我们会在计数函数中获得有趣的信息。我们的目标是传达一些魅力这些“替代”计数职能表现的“替代”评估，并通过几何镜头看着它们通过几何镜头来编织各种组合互惠定理的统一，这将包括通过其他Combinatorial概念的一些景区绕行。
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