Local Limit Theorems for Hook Lengths in Partitions

Abstract

Partition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed \(t\ge 1,\) if \(Y_{t;\,n}\) counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for \(Y_{t;\,n}\) on any bounded set of \({\mathbb {R}}.\) To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define \({\widehat{Y}}_{t;\,n}\) as the count of hooks divisible by t in a randomly chosen partition of n. While \({\widehat{Y}}_{t;\,n}\) converges in distribution, we show that it fails to satisfy the local limit theorem for any \(t \ge 2\). The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for \(t=4,\) the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.