A bedrock of combinatorics, integer partitions underlie many diverse areas of mathematics and physics; the representation theory of the symmetric group and statistical mechanics are two important examples.

They are generated by the $q$-series, $$\prod_{k \geq 1} \frac{1}{1-q^k},$$ which happens to be a modular form, and understanding this modular infinite product has led to a wealth of arithmetic and (asymptotic) statistical information about partitions.

I will first give a brief overview of the classical topics above. Then I will discuss how recent advances in the theory of automorphic forms may be similarly applied to related combinatorial objects.

I will also discuss statistical distributions for representations of uniformly random Lie algebras of large dimension.