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Distribution of hooks in self-conjugate partitions

William Craig, Ken Ono, Ajit Singh

TL;DR

This work establishes that the distribution of the number of $t$-hooks in self-conjugate partitions of size $n$ is asymptotically normal, with explicit leading terms for the mean and variance that depend on $t$ through a parity-based indicator $δ_t$. The authors develop a two-variable generating function $F_t(T;q)$ via the Littlewood bijection and a detailed $q$-series analysis, combining generating-function methods, the dilogarithm, Euler–Maclaurin summation, and saddle-point techniques. They first derive asymptotics for $sc_t(n;T)$ across ranges of $T$, then, through the method of moments, prove the central limit behavior for self-conjugate partitions as $n \to \infty$. A key finding is that the main term of the mean matches the unrestricted-case result while the variance term is doubled in the self-conjugate setting, highlighting how shape constraints alter probabilistic hook statistics. The results deepen connections between partition hook lengths, modular-type $q$-series identities, and probabilistic limits in restricted partition families, with potential implications for related combinatorial and representation-theoretic phenomena.

Abstract

We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate partitions is asymptotically normally distributed with mean $μ_t(n) \sim \frac{\sqrt{6n}}π + \frac{3}{π^2} - \frac{t}{2}+\frac{δ_t}{4}$ and variance $σ_t^2(n) \sim \frac{(π^2 - 6) \sqrt{6n}}{π^3},$ where $δ_t:=1$ if $t$ is odd, and is 0 otherwise.

Distribution of hooks in self-conjugate partitions

TL;DR

This work establishes that the distribution of the number of -hooks in self-conjugate partitions of size is asymptotically normal, with explicit leading terms for the mean and variance that depend on through a parity-based indicator . The authors develop a two-variable generating function via the Littlewood bijection and a detailed -series analysis, combining generating-function methods, the dilogarithm, Euler–Maclaurin summation, and saddle-point techniques. They first derive asymptotics for across ranges of , then, through the method of moments, prove the central limit behavior for self-conjugate partitions as . A key finding is that the main term of the mean matches the unrestricted-case result while the variance term is doubled in the self-conjugate setting, highlighting how shape constraints alter probabilistic hook statistics. The results deepen connections between partition hook lengths, modular-type -series identities, and probabilistic limits in restricted partition families, with potential implications for related combinatorial and representation-theoretic phenomena.

Abstract

We confirm the speculation that the distribution of -hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length among the size self-conjugate partitions is asymptotically normally distributed with mean and variance where if is odd, and is 0 otherwise.
Paper Structure (10 sections, 10 theorems, 82 equations, 2 figures, 1 table)

This paper contains 10 sections, 10 theorems, 82 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and Statement of results
  2. Nuts and Bolts
  3. Generating functions
  4. The dilogarithm function
  5. Euler--Maclaurin summation formulas
  6. Asymptotic lemmas
  7. Asymptotics for $\mathrm{sc}(n;T)$
  8. Proof of Theorem \ref{['T: Main Theorem']}
  9. Finding Mean and Variance
  10. Proof of Theorem \ref{['T: Main Theorem']}

Key Result

Theorem 1.1

For $t \geq 1$ an integer, the function $N_{t,n}$ has an asymptotically normal distribution as $n \to \infty$ with mean asymptotic to $\frac{\sqrt{6n}}{\pi} - \frac{t}{2}$ and variance asymptotic to $\frac{\left( \pi^2 - 6 \right) \sqrt{6n}}{2\pi^3}$.

Figures (2)

  • Figure 1: Hook numbers for the partition $\lambda = \left( 5, 4, 2 \right)$.
  • Figure 2: Renormalized plot of the coefficients of $\mathrm{sc}_2(5000;T)$.

Theorems & Definitions (17)

  • Theorem 1.1: GOT
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Theorem 2.1: AAOS
  • Lemma 2.2: BJM
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more