1 research outputs found
Exact generating function for 2-convex polygons
Polygons are described as almost-convex if their perimeter differs from the
perimeter of their minimum bounding rectangle by twice their `concavity index',
. Such polygons are called \emph{-convex} polygons and are characterised
by having up to indentations in their perimeter. We first describe how we
conjectured the (isotropic) generating function for the case 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 and direction are distinguished. In so doing, we
develop tools that would allow for the case 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.Comment: 28 pages, 15 figures, IoP styl