We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values for perimeter tiles in convex centrally symmetric 2n-gons. These bounds and values have implications for the combinatorics of reduced decompositions of permutations. We also classify the permutations whose polygons gave minimal perimeter, defined in two different ways. We conclude by looking at some of these questions in the context of domino tilings, giving a recursive formula and generating function for one family, and describing a family of minimal-perimeter regions.

中文翻译：

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基于平铺的周长和面积模型

我们考虑某些区域的多边形平铺，并使用这些来给出基于平铺的周长和面积的直观定义。我们将这些定义应用于 Elnitsky 多边形的菱形平铺，计算凸形中心对称 2n 边形中周边平铺的锐界和平均值。这些界限和值对排列的简化分解的组合学有影响。我们还对多边形给出最小周长的排列进行分类，以两种不同的方式定义。最后，我们在多米诺瓷砖的背景下研究其中一些问题，给出一个家庭的递归公式和生成函数，并描述一个最小周长区域的家庭。