Exact generating function for 2-convex polygons
نویسندگان
چکیده
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their ‘concavity index’, m. Such polygons are called m-convex polygons and are characterized by having up to m indentations in their perimeter. We first describe how we conjectured the (isotropic) generating function for the case m = 2 using a numerical procedure based on series expansions. We then proceed to prove this result for the more general case of the full anisotropic generating function, in which steps in the x and y directions are distinguished. In doing so, we develop tools that would allow for the case m > 2 to be studied. PACS numbers: 02.10.Ox, 05.50.+q, 05.70.Jk (Some figures in this article are in colour only in the electronic version)
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