Walks with small steps in the quarter plane

Understand

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, which leads to algebraic generating functions.