Combinatorial Reciprocity Theorems
Combinatorial Reciprocity Theorems
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组合互易定理
DOI：
10.1365/s13291-011-0035-6
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
M. Beck
中科院分区：
文献类型：
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作者：
M. Beck
计数组合学的一个共同主题是通过对在正整数处求值的多项式函数进行计数来形成的。在这篇临时论文中，我们专注于四个家庭这样的计数函数连接到超平面安排，多面体中的格点，正常染色的图，和P-分区。我们将看到，在每一个实例中，当我们在负整数处对计数函数进行求值时，我们都会从计数函数中获得有趣的信息（因此，先验地，计数函数在这个数字处没有意义）。我们的目标是传达一些魅力，这些“替代”的计数功能的评价展览，并编织一个统一的线程通过各种组合互惠定理，通过几何的透镜，这将包括一些风景绕道通过其他组合概念。
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 does not make sense at this number). Our goals are to convey some of the charm these “alternative” evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.