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In the 1970s, Tutte developed a clever algebraic approach, based on certain"invariants" , to solve a functional equation that arises in the enumeration ofproperly colored triangulations. The enumeration of plane lattice walksconfined to the first quadrant is governed by similar equations, and has led inthe past decade to a rich collection of attractive results dealing with thenature (algebraic, D-finite or not) of the associated generating function,depending on the set of allowed steps. We first adapt Tutte's approach to prove(or reprove) the algebraicity of all quadrant models known or conjectured to bealgebraic (with one small exception). This includes Gessel's famous model, andthe first proof ever found for one model with weighted steps. To be applicable,the method requires the existence of two rational functions called invariantand decoupling function respectively. When they exist, algebraicity comes out(almost) automatically. Then, we move to an analytic viewpoint which hasalready proved very powerful, leading in particular to integral expressions ofthe generating function in the non-D-finite cases, as well as to proofs ofnon-D-finiteness. We develop in this context a weaker notion of invariant. Nowall quadrant models have invariants, and for those that have in addition adecoupling function, we obtain integral-free expressions of the generatingfunction, and a proof that this series is differentially algebraic (that is,satisfies a non-linear differential equation).

Counting quadrant walks via Tutte's invariant method (extended abstract)