Exact generating function for 2-convex polygons

Details Exact generating function for 2-convex polygons Exact generating function for 2-convex polygons

2008-05-09

arxiv.org/abs/0805.1259v1

J. Phys. A: Math. Theor. 41 (2008) 055001

Mathematics

Combinatorics

28 pages, 15 figures, IoP style

Scientific paper

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 \emph{$m$-convex} polygons and are characterised 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 $y$ direction are distinguished. In so doing, we develop tools that would allow for the case $m > 2$ to be studied. %In our proof we use a `divide and conquer' approach, factorising 2-convex %polygons by extending a line along the base of its indents. We then use %the inclusion-exclusion principle, the Hadamard product and extensions to %known methods to derive the generating functions for each case.

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