On fully extended self-avoiding polygons (Q1916138)

The authors study self-avoiding polygons in \(\mathbb{Z}^n\) with the conditions (a) fully extended: there are \(2n\) edges in the polygon with their direction vectors spanning \(\mathbb{Z}^n\), and (b) almost fully extended: there are \(2n\) edges in the polygon with their direction vectors spanning a hyperplane of dimension \(n- 1\) in \(\mathbb{Z}^n\). For the numbers of these two kinds of polygons, P. Rossi obtained recurrence formulae in an unpublished preprint. The authors of this note simplify the proofs to these formulae. The enumeration problem for general self-avoiding polygons in \(\mathbb{Z}^n\) is still open.