Enumeration of Almost‐Convex Polygons on the Square Lattice

We classify self‐avoiding polygons on the square lattice according to a concavity measure, m, where 2m is the difference between the number of steps in the polygon and the perimeter of the minimal rectangle bounding the polygon. We generate series expansions for the perimeter generating functions S m (x) for polygons of concavity m. We analyze the series S m (x) for m = 0 to 3. If N m,n denotes the number of polygons of perimeter 2n and concavity m, with m = o (n 1/2 ), we prove that N m,n ˜ 2 2n−m−7 n m+1 /m!, and that the radius of convergence of the series counting all polygons with m = o(n) is equal to 1/4. Our numerical data leads us to conjecture that in fact magnified image for m = o (n 1/2 ), a result confirmed for m = 0 and 1.