We establish, for every family of orthogonal polynomials in the q-Askey scheme and the Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall graph. This generalizes the previous results of Corteel and Kim for the little q-Jacobi polynomials. We build these combinatorial models by bootstrapping, beginning with polynomials at the bottom and working towards Askey-Wilson polynomials which sit at the top of the q-Askey scheme. As an application of the theory, we provide the first combinatorial proof of the symmetries in the parameters of the Askey-Wilson polynomials.

Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded tableaux thanks to two other combinatorial models: non intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octogons. We use the tangent method to investigate the arctic curve in the model of nonintersecting lattice paths with fixed starting points and ending points distributibuted according to some arbitrary piecewise differentiable function. We then study the dimer model and use some ansatz to guess the asymptotics of the inverse of the Kasteleyn matrix confirm the arctic curve computed with the tangent method for two examples.