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Inequalities and asymptotics for hook numbers in restricted partitions

William Craig, Madeline Locus Dawsey, Guo-Niu Han

TL;DR

The paper studies hook-length statistics in restricted partitions, comparing partitions into odd parts with partitions into distinct parts. It develops explicit generating functions for the hook-counts $a_h(n)$ and $b_h(n)$, showing they factor as a modular form factor $(-q;q)_\infty$ times rational pieces, and then derives asymptotics for large $n$ via Euler--Maclaurin summation and Wright's circle method. It proves that for every $h\ge2$, $a_h(n)/b_h(n) \to \gamma_h>1$, and that $\gamma_h \to \log 4/\log 3$ as $h\to\infty$, thereby confirming the BBCFW conjecture in full. The work also provides detailed probabilistic corollaries about how hooks distribute across rows and parts, yielding precise growth and distribution formulas and highlighting striking differences between the two partition families. The results connect hook-length phenomena to modular forms and offer avenues for generalizing to other partition statistics and related combinatorial objects.

Abstract

In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers equal to $h \geq 1$ by deriving an asymptotic formula for the total number of hooks equal to $h$ that appear among partitions into odd and distinct parts, respectively. We use these asymptotic formulas to prove a recent conjecture of the first author and collaborators that for $h \geq 2$ and $n \gg 0$, partitions into odd parts have, on average, more hooks equal to $h$ than do partitions into distinct parts. We also use our asymptotics to prove certain probabilistic statements about how hooks distribute in the rows of partitions.

Inequalities and asymptotics for hook numbers in restricted partitions

TL;DR

The paper studies hook-length statistics in restricted partitions, comparing partitions into odd parts with partitions into distinct parts. It develops explicit generating functions for the hook-counts and , showing they factor as a modular form factor times rational pieces, and then derives asymptotics for large via Euler--Maclaurin summation and Wright's circle method. It proves that for every , , and that as , thereby confirming the BBCFW conjecture in full. The work also provides detailed probabilistic corollaries about how hooks distribute across rows and parts, yielding precise growth and distribution formulas and highlighting striking differences between the two partition families. The results connect hook-length phenomena to modular forms and offer avenues for generalizing to other partition statistics and related combinatorial objects.

Abstract

In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers equal to by deriving an asymptotic formula for the total number of hooks equal to that appear among partitions into odd and distinct parts, respectively. We use these asymptotic formulas to prove a recent conjecture of the first author and collaborators that for and , partitions into odd parts have, on average, more hooks equal to than do partitions into distinct parts. We also use our asymptotics to prove certain probabilistic statements about how hooks distribute in the rows of partitions.
Paper Structure (10 sections, 22 theorems, 137 equations, 3 figures)

This paper contains 10 sections, 22 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generating Functions
  3. Preliminaries
  4. Euler--Maclaurin Summation
  5. Wright's circle method
  6. Asymptotics for $a_h(n)$ and $b_h(n)$
  7. Evaluating $\alpha_h$ and $\beta_h$
  8. Final Remarks
  9. Frequency of hooks in the rows of partitions
  10. Open problems and questions

Key Result

Theorem 1.2

Conjecture BBCFW Conjecture (2), and therefore also Conjecture BBCFW Conjecture (1), is true for all $h \geq 2$.

Figures (3)

  • Figure 1: Hook numbers of the partition $\lambda=(4,3,2)$
  • Figure 2: Arm, leg, coarm, and coleg lengths: $\mathop{\mathrm{arm}}\nolimits(\lambda, v)=j$, $\mathop{\mathrm{leg}}\nolimits(\lambda, v)=l$, $\mathop{\mathrm{coarm}}\nolimits(\lambda, v)=m$, $\mathop{\mathrm{coleg}}\nolimits(\lambda, v)=g$
  • Figure 3: A partition and its different regions

Theorems & Definitions (45)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 35 more