Self-avoiding polygons on a three-row square-lattice strip

Abstract

We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\mathbb{Z}\times\{0,1,2\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new geometric proof of an identity arising in Larsen's treatment \cite{L07} of a problem posed by Gessel \cite{G95}. Finally, we show that this subtype of SAPs is closely connected to the sequence A007909. More precisely, for $m\geq0$, the number of these SAPs whose leftmost and rightmost columns each contain two vertical steps and whose length equals $2m+6$ is given by the term of this sequence with index $m$, which thereby acquires a geometric interpretation alongside the compositions it enumerates.