User:StevenCollazos/sandbox

Examples of Ehrhart Quasi-Polynomials
Let P be a polygon with vertices (0,0), (0,2), (1,1) and (0,3/2). The number of integer points in tP will be counted by the quasi-polynomial

$$ L(P, t) = \frac{7}{4}t^2 + \frac{5}{2}t + \frac{7 + (-1)^t}{8}. $$

Generalizations
One natural question to ask is how many integer points there are if we dilate some facets of a polytope but not others. In other words, one would like to know the number of integer points in a semi-dilated polytope. It turns out that such a counting function will be what is called a multivariate quasi-polynomial. A Ehrhart-type reciprocity theorem will also hold for such a counting function.

Counting the number of integer points in semi-dilations of polytopes has applications in enumerating the number of different dissections of regular polygons and the number of unrestricted codes, a particular kind of code.