Maximum entropy and integer partitions

TL;DR

This paper develops asymptotic formulas for counting integer partitions with specific power sum constraints using the maximum entropy principle, framing the problem as a convex optimization task.

Contribution

It introduces a novel application of the maximum entropy principle to derive asymptotic formulas for constrained integer partitions.

Findings

Asymptotic formulas for partition counts with power sum constraints.

Variational formula based on maximum entropy principle.

Convex optimization approach to asymptotic enumeration.

Abstract

We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.