W R G James et al 2008 J. Phys. A: Math. Theor. 41 055001 doi:10.1088/1751-8113/41/5/055001
Exact generating function for 2-convex polygons
W R G James, I Jensen and A J Guttmann
Show affiliationsPolygons 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
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02.10.Ox Combinatorics; graph theory
- MSC
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05A15 Exact enumeration problems, generating functions (See also 33Cxx, 33Dxx)
41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
- Subjects
- Dates
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Issue 5 (8 February 2008)
Received 25 May 2007, in final form 1 November 2007
Published 23 January 2008
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Exact generating function for 2-convex polygons
W R G James et al 2008 J. Phys. A: Math. Theor. 41 055001
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