3D positive lattice walks and spherical triangles Abstract   In this paper we explore the asymptotic enumeration of threedimensional excursions confined to the positive octant. As shown in  29 , both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the socalled group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closedform formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finiteelements method to compute approximate values, typically with ten digits of precision.

The steepbounce zeta map in parabolic Cataland Abstract   As a classical object, the Tamari lattice has many generalizations, including xcexbdTamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to xcexbdTamari lattices for bounce paths xcexbd. We then introduce a new combinatorial object called xe2x80x9cleftaligned colorable treexe2x80x9d, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the SteepBounce Conjecture using a generalization of the famous zeta map in    q  ,  t   Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.