Asymptotic Statistics of Odd Unimodal Sequences: Rank Distributions and Probabilistic Structures

TL;DR

This paper develops a comprehensive asymptotic statistical theory for odd unimodal sequences, revealing their rank distribution converges to a hyperbolic secant distribution and exploring their probabilistic structure.

Contribution

It extends the probabilistic analysis of unimodal sequences to the odd-part constrained case, combining modular forms, false theta asymptotics, and Boltzmann models.

Findings

Rank distribution converges to hyperbolic secant distribution.

Limiting distributions of peak and largest parts are established.

The probabilistic structure exhibits distinctive features due to the odd-part constraint.

Abstract

Integer partitions have fascinated people for centuries, from Ramanujan's groundbreaking congruences to the modern theory of modular forms. This paper investigates the statistical properties of odd unimodal sequences--a natural refinement where sequences rise to a peak and then fall, but with the constraint that all parts must be odd, and develops a comprehensive statistical theory for their rank and shape parameters. We establish the asymptotic distribution of the rank statistic and demonstrate that, when properly normalized, it converges to the hyperbolic secant distribution. Beyond the rank distribution, limiting distributions of the peak, the largest parts on either side of the peak, and the joint behavior of small parts are also proved. These results reveal a rich probabilistic structure that parallels the classical theory of integer partitions while exhibiting distinctive new features arising from the odd-part constraint. The analysis employs a synthesis of modular transformation theory, false theta function asymptotics, and conditioned Boltzmann models. This extends the probabilistic machinery previously developed for unimodal sequences into a more general and analytically demanding setting, offering a unified approach that bridges modular forms and probability.