Abstract
AbstractPartition 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 $$t\ge 1,$$ t ≥ 1 , if $$Y_{t;\,n}$$ 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 $$Y_{t;\,n}$$ Y t ; n on any bounded set of $${\mathbb {R}}.$$ 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 $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n as the count of hooks divisible by t in a randomly chosen partition of n. While $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n converges in distribution, we show that it fails to satisfy the local limit theorem for any $$t \ge 2$$ t ≥ 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 $$t=4,$$ t = 4 , the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.
Published Version
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