math.MP — Mathematical Physics

Riordan lattice paths counted by area

Contributed by Ilmar Gahramanov

We study Riordan paths enumerated by the area between the path and the axis. We show that the generating function satisfies a $q$-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 $n$ we obtain a joint Gaussian limit law for the numbers of returns, horizontal steps and up steps. The height is of order $\sqrt{\log n}$, and its distribution concentrates on two adjacent integers determined explicitly by $n$.