Counting integer partitions with the method of maximum entropy

Abstract: We give an asymptotic formula for the number of partitions of an integer $n$ where the sums of the $k$th powers of the parts are also fixed, for some collection of values $k$. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.

A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

combinatorics

Audience: researchers in the topic

Warwick Combinatorics Seminar

Series comments: This is the online combinatorics seminar at Warwick.