Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the distribution of $t$-hooks of doubled distinct partitions

Hyunsoo Cho, Byungchan Kim, Eunmi Kim, Ae Ja Yee

TL;DR

The paper studies the distribution of $t$-hooks in doubled distinct partitions and of $t$-shifted hooks in strict partitions by deriving explicit generating functions via the Littlewood decomposition. Using a circle-method framework, it proves that both $N_{t,2n}^{\mathcal{DD}}$ and $\widehat{N}_{t,2n}^{\mathcal{DD}}$ are asymptotically normal, and it provides detailed asymptotic formulas for their means and variances. The results extend prior work on hook-length distributions from all partitions and self-conjugate partitions to these two structured families, linking combinatorial decompositions with analytic techniques such as the dilogarithm and Wright’s circle method. The generating functions $F_t(x;q)$ and $\widehat{F}_t(x;q)$ encode the joint structure of $t$-cores and $t$-quotients, enabling precise asymptotics and moment calculations. Overall, the work sheds light on universal normality phenomena in partition hook-length statistics and broadens the landscape of partition-analytic methods in representation-theoretic contexts.

Abstract

Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate partitions of $n$. Motivated by these studies, in this paper, we further investigate the number of $t$-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions are both asymptotically normally distributed.

On the distribution of $t$-hooks of doubled distinct partitions

TL;DR

The paper studies the distribution of -hooks in doubled distinct partitions and of -shifted hooks in strict partitions by deriving explicit generating functions via the Littlewood decomposition. Using a circle-method framework, it proves that both and are asymptotically normal, and it provides detailed asymptotic formulas for their means and variances. The results extend prior work on hook-length distributions from all partitions and self-conjugate partitions to these two structured families, linking combinatorial decompositions with analytic techniques such as the dilogarithm and Wright’s circle method. The generating functions and encode the joint structure of -cores and -quotients, enabling precise asymptotics and moment calculations. Overall, the work sheds light on universal normality phenomena in partition hook-length statistics and broadens the landscape of partition-analytic methods in representation-theoretic contexts.

Abstract

Recently, Griffin, Ono, and Tsai examined the distribution of the number of -hooks in partitions of , which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of -hooks in self-conjugate partitions of . Motivated by these studies, in this paper, we further investigate the number of -hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of -hooks in doubled distinct partitions and the number of -shifted hooks in strict partitions. Based on these generating functions, we prove that the number of -hooks in doubled distinct partitions and the number of -shifted hooks in strict partitions are both asymptotically normally distributed.

Paper Structure

This paper contains 18 sections, 19 theorems, 141 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. -Series
  4. Durfee square and Frobenius symbol
  5. Wright's map
  6. Littlewood decomposition
  7. Analytic properties
  8. Generating functions
  9. Littlewood decomposition for doubled distinct partitions
  10. The generating function
  11. The generating function
  12. Asymptotics of the generating functions
  13. Asymptotic behavior of for odd
  14. Asymptotic behavior of for even
  15. Asymptotics of
  16. ...and 3 more sections

Key Result

Theorem 1.1

For a positive integer $t$, where

Figures (5)

  • Figure 1: The Young diagram of the partition $(5,4,1)$ with hook lengths
  • Figure 2: The shifted Young diagram of the partition $(5,4,1)$ with its shifted hook lengths
  • Figure 3: The Young diagram of the partition $(6,6,4,2,2)$ with its hook lengths
  • Figure 4: The distribution of $N_{3, n}^{\mathcal{DD}}$
  • Figure 5: Two doubled distinct partitions in $\mathcal{DD}_{(1)}$ and $\mathcal{DD}_{(2)}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Proposition 3.2
  • ...and 25 more