Exactly Solved Models

This chapter deals with the exact enumeration of certain classes of (self-avoiding) polygons and polyominoes. We restrict our attention to the square lattice. As the interior of a polygon is a polyomino, we often consider polygons as special poly-ominoes. The usual enumeration parameters are the area (the number of cells) and the perimeter (the length of the border). The perimeter is always even, and often refined into the horizontal and vertical perimeters (number of horizontal/vertical steps in the border). Given a class C of polyominoes, the objective is to determine the following complete generating function of C:

$$C\left( {x,y,q} \right) = \sum\limits_{P \in d} {x^{hp\left( P \right)/2_y vp\left( P \right)/2_q a\left( P \right)},} $$

where hp(P), vp(P) and a(P) respectively denote the horizontal perimeter, the vertical perimeter and the area of P. This means that the coefficient c(m,n,k) of x ^m y ⁿ q ^k in the series C(x,y,q) is the number of polyominoes in the class C having horizontal perimeter 2m, vertical perimeter 2n and area k. Several specializations of C(x,y,q) may be of interest, such as the perimeter generating function C(t,t,1), its anisotropic version C(x,y,1), or the area generating function C(1,1,q). From such exact results, one can usually derive many of the asymptotic properties of the poly-ominoes of C : for instance the asymptotic number of polyominoes of perimeter n, or the (asymptotic) average area of these polyominoes, or even the limiting distribution of this area, as n tends to infinity (see Chapter 11). The techniques that are used to derive asymptotic results from exact ones are often based on complex analysis. A remarkable survey of these techniques is provided by Flajolet and Sedgewick's book [33].