A half-normal distribution scheme for generating functions and the unexpected behavior of Motzkin paths

TL;DR

This paper extends a theorem to identify limiting distributions of combinatorial structures, revealing a half-normal distribution in Motzkin paths, which broadens understanding of their probabilistic behavior.

Contribution

It introduces an extension of existing theorems to determine when a half-normal distribution arises in combinatorial generating functions, especially applied to Motzkin paths.

Findings

Identifies conditions for half-normal limiting distribution

Extends previous theorems to broader classes of distributions

Demonstrates the distribution in Motzkin path applications

Abstract

We present an extension of a theorem by Michael Drmota and Mich\`ele Soria [Images and Preimages in Random Mappings, 1997] that can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytical and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of Motzkin paths.