Average site perimeter of directed animals on the two-dimensional lattices

Axel Bacher (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le~Borgne. The techniques used are based on Viennot's correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).

Mots clés: directed animal; enumerative combinatorics; generating function

Domaines: Combinatoire [math.CO]

Axel Bacher. Average site perimeter of directed animals on the two-dimensional lattices. Discrete Mathematics, 2012, 312 (5), pp.Pages 1038-1058. ⟨10.1016/j.disc.2011.11.008⟩. ⟨hal-00398436v3⟩