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Section: New Results

Non-D-finite excursions in the quarter plane

Counting lattice paths obeying various geometric constraints is a classical topic in combinatorics and probability theory. Many recent works deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. A large part of the effort has been devoted to the classification of classes of walks according to the nature of equations that they satisfy (linear, polynomial, differential, etc). Equivalently, this provides properties of the classes of walks according to the algebraic nature of their enumerative series: whether rational, algebraic, D-finite, etc. The classification is now complete for walks with unit steps: the trivariate generating function of the numbers of walks with given length and prescribed ending point is D-finite if and only if a certain group associated with the step set is finite. We proved in [5] a refinement of this result: we showed that the sequence of numbers of excursions (finite paths starting and ending at the origin) in the quarter plane corresponding to a nonsingular step set with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. This solves an open problem in the field of lattice-path combinatorics.