Combinatorial Theory

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. Finding the polynomial corrections and leading constant involves two steps: quantifying the error in approximating a discrete optimization problem by a continuous one and proving a multivariate local central limit theorem.

Mathematics Subject Classifications: 05A17, 05A16, 60F05

Keywords: Integer partitions, maximum entropy, asymptotic enumeration, local central limit theorem, limit shape