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Maximum entropy and integer partitions

Published Web Location Commons 'BY' version 4.0 license

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

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