Peak positions of strongly unimodal sequences

TL;DR

This paper investigates the combinatorial and asymptotic properties of the rank in strongly unimodal sequences, providing new generating functions, interpretations, and asymptotic normality results.

Contribution

It introduces a new generating function for the rank enumeration, offers a novel combinatorial interpretation of the ospt-function, and proves the asymptotic normality of the rank.

Findings

Generated a new rank enumeration function.

Provided a combinatorial interpretation of the ospt-function.

Proved asymptotic normal distribution of the rank.

Abstract

We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function, and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and prove an asymptotic result in support of this conjecture. Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and prove that its values (when appropriately renormalized) are normally distributed with mean zero in the asymptotic limit.