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Riordan lattice paths counted by area

A mix of human-written and AI-generated textHuman understanding: some partsmath.MP — Mathematical Physicsmath.PR — Probability

Contributed by Ilmar Gahramanov ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

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

Provenance statement

Large language models were used in the literature search, computer-assisted checks and preparation of the manuscript. The original derivation of Section~\ref{sec:limit} and Appendix~\ref{app:proof} was obtained entirely with the assistance of Claude (Anthropic). ChatGPT (OpenAI) assisted with subsequent mathematical and editorial refinements, including the joint Gaussian and two-height corollaries, and with independent numerical checks.

Tools used

Anthropic
Claude Fable
OpenAI
ChatGPT

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Version history

  1. v1Submitted by Ilmar GahramanovInitial depositCurrentOct 31, 2026