The way I interpreted the original question, based on Allan's description and his proposed sequence 1, 1, 2, 5, 10, 25 was: Given a polyomino P on a square lattice, if you replace each of the squares in P with a point (say the "upper-left" corner) and call that set of points S, and then define H to be the convex hull of S, then the polyomino is said to be "disklike" if all lattice points in H are also in S. Since my previous message I took a nap and then decided to give up figuring out the whole thing myself. Since I am taking a convex hull of points aligned on a lattice the algorithm is a lot simpler than the general case, it seems I need little more than a test for whether an angle ABC is less than or greater than 180 degrees (e.g. the determinant formula for the signed area of a triangle, see http://mathworld.wolfram.com/TriangleArea.html formulas 16 and 17). I can start by confirming that each "row" of the polyomino is a single contiguous run of squares (rejecting the U pentomino in "horizontal" orientation, and all polyominoes with holes A001419). Then I can use the left and right ends of each row to define a set of points which is a superset of the convex hull, and the ABC angle test (walking up the left side of the polyomino, and walking down the right side) to eliminate points that are not one of the "vertices" of the convex hull. Finally I can look at the lattice points closest to the extreme ends of each row (except the first and last) to determine if they are within the hull. That test would reject the V pentomino (whose convex hull is a right triangle), on the basis that its middle row contains a single lattice point, one of whose neighbors falls on the hypotenuse. On Sat, May 7, 2011 at 13:38, Dan Asimov <dasimov@earthlink.net> wrote:
Given a closed & bounded convex figure K:
Does "intersection between any convex figure and the lattice" mean the union of grid squares that intersect K ?
Or the union of grid squares that are each entirely contained in K ?
(Or maybe both definitions yield the same set of polyominoes?)
* * *
Allan's definitions of disklike polyomino, and the one below, suggest another one:
* A polyomino that is topologically a closed disk.
[...]
Allan wrote:
<< On a previous occasion, I talked about enumerating "disklike" polyominoes, polyominoes which occurred as the intersection between a disk and a square lattice. Today I tried counting polyominoes which occurred as the intersection between any convex figure and the lattice. This implies that the convex hull of the polyomino (viewed as a set of lattice points) includes no additional lattice points.
Sometimes the brain has a mind of its own.
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