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Enumeration of Almost‐Convex Polygons on the Square Lattice
Author(s) -
Enting I. G.,
Guttmann A. J.,
Richmond L. B.,
Wormald N. C.
Publication year - 1992
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.3240030407
Subject(s) - perimeter , combinatorics , mathematics , polygon (computer graphics) , rectangle , regular polygon , star shaped polygon , rectilinear polygon , square lattice , series (stratigraphy) , conjecture , point in polygon , enumeration , polygon covering , square (algebra) , lattice (music) , geometry , simple polygon , convex set , physics , telecommunications , statistical physics , paleontology , convex optimization , frame (networking) , computer science , acoustics , ising model , biology
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 factfor m = o (n 1/2 ), a result confirmed for m = 0 and 1.