Counting quadrant walks via Tutte's invariant method

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve functional equation that arises in enumeration of properly colored triangulations. The plane lattice walks confined first quadrant is governed by similar equations, and has led past 20 years rich collection attractive results dealing with nature (algebraic, D-finite or not) associated generating function, depending set allowed steps, taken \(\{-1, 0,1\}^2\).We adapt Tutte's approach prove (or reprove) algebraicity all models known conjectured be algebraic. This includes Gessel's famous model, proof ever found for one model weighted steps. To applicable, method requires existence two rational functions called invariant decoupling function respectively. When they exist, follows almost automatically.Then, we move complex analytic viewpoint already proved very powerful, leading particular integral expressions non-D-finite cases, as well proofs non-D-finiteness. We develop this context weaker notion invariant. Now have invariants, those addition obtain integral-free series D-algebraic (that is, satisfies polynomial differential equations).Keywords: Lattice walks, enumeration, differentially series, conformal mappings.Mathematics Subject Classifications: 05A15, 34K06, 39A06, 30C20, 30D05