Exact solution of the row-convex polygon perimeter generating function

Abstract

An explicit expression is obtained for the perimeter generating function G(y)= Sigma _n>or=2 a_ny²ⁿ for row-convex polygons on the square lattice, where a_n is the number of 2n step row-convex polygons. An asymptotic expression for a_n approximately A mu ⁿn^-3/2 is obtained, where mu =3+22 and A are given. The authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three.