Limit shapes via bijections

TL;DR

This paper computes the limit shapes of restricted integer partitions using bijections that map these shapes continuously, extending previous methods to include various scaling functions.

Contribution

It introduces an extended framework of bijections to analyze limit shapes of partitions with size restrictions, broadening prior approaches.

Findings

Derived explicit limit shapes for various restricted partitions.

Extended bijection methods to include different scaling functions.

Provided a unified approach to analyze partition limit shapes.

Abstract

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined previously by the second author, and extend them to include limit shapes with different scaling functions.