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Hook length biases for self-conjugate partitions and partitions with distinct odd parts

Catherine Cossaboom

Abstract

We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length $t \geq 2$ among self-conjugate partitions of $n$ than among partitions of distinct odd parts of $n$ for sufficiently large $n$. More precisely, we derive asymptotic formulas for the total number of hooks of fixed length $t$ in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.

Hook length biases for self-conjugate partitions and partitions with distinct odd parts

Abstract

We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length among self-conjugate partitions of than among partitions of distinct odd parts of for sufficiently large . More precisely, we derive asymptotic formulas for the total number of hooks of fixed length in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.

Paper Structure

This paper contains 18 sections, 23 theorems, 123 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Generating Functions for $a^*_t(n)$ and $b^*_t(n)$
  3. Asymptotics for $A^*_t(q)$ and $B^*_t(q)$
  4. Proof of Proposition \ref{['Atasymptotics']}
  5. Proof of Proposition \ref{['Btasymptotics']}
  6. Asymptotics for $a^*_t(n)$ and $b^*_t(n)$
  7. A Variation of Wright's Circle Method
  8. Major and Minor Arc Computations for $K_t(q)$
  9. Major and Minor Arc Computations for $L_t(q)$
  10. Major and Minor Arc Computations for $\xi(q)$
  11. Proof of Theorem \ref{['theorem2']}(1) and (2)
  12. A Probabilistic Consequence
  13. Evaluating $\beta^*_t$
  14. Simplifying $S_{r,s}(t)$
  15. A Combinatorial Interlude
  16. ...and 3 more sections

Key Result

Theorem 1.2

Conjecture conj is true.

Figures (4)

  • Figure 1: Hook numbers for the partition (5,3,2)
  • Figure 2: Values of $\gamma_t^*$ for various $t$
  • Figure 3: Arm, coarm, leg, and coleg length of $\lambda = (10,9,9,9,8,5,1,1)$
  • Figure 4: Regions $A$, $B$, $C$, and $D$ of $\lambda = (10,9,9,9,8,5,1,1)$

Theorems & Definitions (35)

  • Conjecture 1.1: Ballantine--Burson--Craig--Folsom--Wen, Craig--Dawsey--Han
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: AAOS
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 25 more