Maximum entropy and integer partitions

Abstract

We derive asymptotic formulas for the number of integer partitions with given sums of jth powers of the parts for j belonging to a finite, non-empty set J ⊂ 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.

Independent sets of a given size and structure in the hypercube

“…It follows that to show (17) holds whp with respect to Γ, it suffices to show that 1) , (18) and similarly for |N (Γ)|. This is an immediate consequence of Lemma 11 and the fact that…”

“…There is a long history of using local central limit theorems in combinatorics, with many examples in analytic combinatorics and the study of integer partitions (see e.g. [19,7,18,17]) as well as the enumeration of contingency tables [4] and graphs with prescribed degree sequences (e.g. [5,12]).…”

“…The connection between these methods starts with a general approach to counting via probability and the principle of maximum entropy which is laid out explicitly by Barvinok and Hartigan in [3] (and later discussed in [17]), but appears implicitly in other enumerations methods, such as the circle method (see [12] for an explanation of these connections). The main idea is that to count a subset of objects defined by a number of constraints, one considers the maximum entropy distribution on the larger set that satisfies the constraints in expectation.…”

“…The size of the subset can then be expressed as the exponential of the entropy of this distribution times the probability that a random object drawn from this distribution satisfies the constraints. In the example of enumerating integer partitions, these maximum entropy distributions take the form of sequences of independent geometric random variables with different means [9,1], and asymptotic enumeration can be accomplished by solving a convex optimization problem to find these means and then proving a local central limit theorem for linear combinations of independent geometric random variables [19,7,18,17]. This approach naturally leads to considering statistical physics models.…”