The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding Polygons and Walks

Abstract

This is the fourth installment of the five-part saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota's seminal notion of the umbra. In this article we describe the Maple packages USAP, USAW, and MAYLIS. USAP automatically constructs, for any specific $r$, an Umbral Scheme for enumerating, according to perimeter, the number of self-avoiding polygons with $\leq 2r$ horizontal edges per vertical cross-section. The much more complicated USAW does the analogous thing for self-avoiding walks. Such Umbral Schemes enable counting these classes of self-avoiding polygons and walks in polynomial time as opposed to the exponential time that is required by naive counting. Finally MAYLIS is targeted to the special case of enumerating classes of saps with at most two horizontal edges per vertical cross-section (equivalently column-convex polyominoes by perimeter), and related classes. In this computationally trivial case we can actually automatically solve the equations that were automatically generated by USAP. As an example, we give the first fully computer-generated proof of the celebrated Delest-Viennot result that the number of convex polyominoes with perimeter $2n+8$ equals $(2n+11)4^n-4(2n+1)!/n!^2$.