math.MP — Mathematical Physics
Riordan lattice paths counted by area
We study Riordan paths enumerated by the area between the path and the axis. We show that the generating function satisfies a -Riccati equation and can be written in a closed form as a ratio of basic hypergeometric series. The auxiliary series admits an algebraic representation as the motivic generating series of a symmetric two-node quiver. We obtain explicit formulas for the coefficients, the exponential asymptotics of the number of paths of given area, and a linear recurrence whose coefficients are signed counts of partitions into distinct parts. For a random path of area we obtain a joint Gaussian limit law for the numbers of returns, horizontal steps and up steps. The height is of order , and its distribution concentrates on two adjacent integers determined explicitly by .