Limit Shapes for Unimodal Sequences

TL;DR

This paper establishes the existence of a unique limit shape for unimodal sequences of positive integers, extending known results from integer partitions and including a limit shape for overpartitions.

Contribution

It proves asymptotic 0-1 laws for unimodal sequence diagrams and identifies their limit shapes, extending classical partition results to new combinatorial structures.

Findings

Almost all shapes are near a specific curve as size grows

Limit shape results extend to overpartitions

Provides a natural extension of partition limit shape phenomena

Abstract

We prove asymptotic 0-1 Laws satisfied by diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane, and the upper boundary is called the shape. For various types, we show that, as the number of squares tends to infinity, $100\%$ of shapes are near a certain curve---that is, there is a single {\it limit shape}. Similar phenomena have been well-studied for integer partitions, so the present work is a natural extension. One notable corollary is a transferred limit shape for overpartitions.