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 $$t\ge 1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> if $$Y_{t;\,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> counts the number of hooks of length t in a random integer partition of n , we prove a uniform local limit theorem for $$Y_{t;\,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> on any bounded set of $${\mathbb {R}}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> 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 $${\widehat{Y}}_{t;\,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> as the count of hooks divisible by t in a randomly chosen partition of n . While $${\widehat{Y}}_{t;\,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> converges in distribution, we show that it fails to satisfy the local limit theorem for any $$t \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 $$t=4,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.