Lattice paths of slope 2/5

We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5. This answers to Knuth’s problem #4 from his “Flajolet lecture” during the conference “Analysis of Algorithms” (AofA’2014) in Paris in June 2014. Our approach relies on the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths. A key ingredient in the proof is the generalization of an old trick of Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the “kernel method”. All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can be also tackled in the A=B spirit of Wilf–Zeilberger–Petkovšek. We show how to obtain similar results for other slopes than 2/5, an interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon’s