Walks with small steps in the quarter plane
Résumé
Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there are 2^8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, and are known to have algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases. We give evidence that it is infinite in the 56 other cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. Our approach allows us to recover and refine some known results, and also obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.
Domaines
Combinatoire [math.CO]
Origine : Fichiers produits par l'(les) auteur(s)