Tiling-based models of perimeter and area

Abstract

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.