W R G James , I Jensen and A J Guttmann
ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
william.james@axa.com I.Jensen@ms.unimelb.edu.au T.Guttmann@ms.unimelb.edu.au
Journal of Physics A: Mathematical and Theoretical Create an alert RSS this journal
W R G James et al 2008 J. Phys. A: Math. Theor. 41 055001
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.
02.10.Ox Combinatorics; graph theory
05A15 Exact enumeration problems, generating functions (See also 33Cxx, 33Dxx)
41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Issue 5 ( 8 February 2008)
Received 25 May 2007
,
in final form 1 November 2007
Published 23 January 2008
W R G James et al 2008 J. Phys. A: Math. Theor. 41 055001
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