The Asymptotic Distribution of the Rank for Unimodal Sequences

TL;DR

This paper investigates the asymptotic distribution of the rank statistic for unimodal sequences, showing it converges to a logistic distribution after normalization, with extensions to related sequence types.

Contribution

It provides new asymptotic formulas for the moments of the unimodal rank and establishes its limiting distribution, including special cases like semistrict sequences.

Findings

Normalized unimodal rank follows a logistic distribution asymptotically

Derived asymptotic formulas for moments of the rank

Identified distributional limits for different unimodal sequence types

Abstract

We study the asymptotic behavior of the rank statistic for unimodal sequences. We use analytic techniques involving asymptotic expansions in order to prove asymptotic formulas for the moments of the rank. Furthermore, when appropriately normalized, the values of the unimodal rank asymptotically follow a logistic distribution. We also prove similar results for Durfee unimodal sequences and semi-strict unimodal sequences, with the only major difference being that the (normalized) rank for semistrict unimodal sequences has a distributional limit of a point mass probability distribution.