Abstract
We provide exact and asymptotic counting formulas for five singular lattice path models in the quarter plane. Furthermore, we prove that these models have a non D-finite generating function. Nous présentons des résultats énumératifs pour les cinq modèles de marche dans le quart de plan dites "singulière''. Nous prouvons que ces modèles sont non-holonomes.
Highlights
The study of lattice path models restricted to the quarter plane has led to some useful innovations in enumeration, including applications of boundary value methods [8, 9, 12], powerful and widely applicable variants of the kernel method [5, 7, 14] and some original computer algebra approaches [3]
A key observation of Bousquet-Melou and Mishna was that lattice path models with small steps restricted to the quarter plane appeared to be naturally partitioned according to the nature of their generating functions: they conjectured a test for whether or not the generating function of a given model would satisfy algebraic or linear differential equations
One can show that the singularities are not canceled in the two summands of (11) containing YnA and YnC so that, as they lie off the unit circle where the remaining summands are analytic by Proposition 5, they give an infinite number of singularities of A1,0, B1,0, and C1,0
Summary
The study of lattice path models restricted to the quarter plane has led to some useful innovations in enumeration, including applications of boundary value methods [8, 9, 12], powerful and widely applicable variants of the kernel method [5, 7, 14] and some original computer algebra approaches [3]. A key observation of Bousquet-Melou and Mishna was that lattice path models with small steps restricted to the quarter plane appeared to be naturally partitioned according to the nature of their generating functions: they conjectured a test for whether or not the generating function of a given model would satisfy algebraic or linear differential equations. This property is often correlated to other, more combinatorial, qualities of a model. This extended abstract presents the main results, and we we refer the reader to [13] for a completed manuscript
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