Statistics for Unimodal Sequences

TL;DR

This paper develops a probabilistic framework to derive limiting distributions for statistics of unimodal sequences, overcoming challenges posed by their generating functions, and extends results to strongly unimodal sequences.

Contribution

It introduces a novel probabilistic approach for unimodal sequences, enabling the derivation of various distributional results including joint distributions and ranks.

Findings

Derived limiting distributions for unimodal sequences

Extended results to strongly unimodal sequences

Provided joint distributions for largest parts and small part multiplicities

Abstract

We prove a number of limiting distributions for statistics for unimodal sequences of positive integers by adapting a probabilistic framework for integer partitions introduced by Fristedt. The difficulty in applying the direct analogue of Fristedt's techniques to unimodal sequences lies in the fact that the generating function for partitions is an infinite product, while that of unimodal sequences is not. Essentially, we get around this by conditioning on the size of the largest part and working uniformly on contributing summands. Our framework may be used to derive many distributions, and our results include joint distributions for largest parts and multiplicities of small parts. We discuss ranks as well. We further obtain analogous results for strongly unimodal sequences.