Statistical structure of concave compositions

TL;DR

This paper investigates the statistical properties of concave compositions, extending partition theory, and provides solutions to open problems about their distribution, shape, and structure using advanced probabilistic methods.

Contribution

It offers the first comprehensive analysis of the statistical structure of concave compositions, solving several open problems with Fristedt's conditioning device.

Findings

Distribution of perimeter and tilt characterized

Number of summands analyzed and described

Shape of typical concave compositions determined

Abstract

In this paper, we study concave compositions, an extension of partitions that were considered by Andrews, Rhoades, and Zwegers. They presented several open problems regarding the statistical structure of concave compositions including the distribution of the perimeter and tilt, the number of summands, and the shape of the graph of a typical concave composition. We present solutions to these problems by applying Fristedt's conditioning device on the uniform measure.