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Hook Length Biases in $t$-Core Partitions

Nayandeep Deka Baruah, Hirakjyoti Das, Pankaj Jyoti Mahanta, Manjil P. Saikia

Abstract

Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in $\ell$-regular partitions. In this work, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n)\ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n)\ge a_{4,3}(n)$ for all $n$. The methods employed in this work are mainly combinatorial.

Hook Length Biases in $t$-Core Partitions

Abstract

Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in -regular partitions. In this work, we extend the theory of hook length biases to -core partitions. For example, let denote the number of hooks of length in all -core partitions of , then we find that and for all . The methods employed in this work are mainly combinatorial.
Paper Structure (5 sections, 9 theorems, 67 equations, 8 figures)

This paper contains 5 sections, 9 theorems, 67 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['Thm 2 Core Formula']}
  3. Combinatorial Proof of Theorem \ref{['Thm HLB 2-4']}
  4. Combinatorial proof of Theorem \ref{['Thm HLB 4 Core 1-3']} and Other Results
  5. Proof of Theorem \ref{['Thm HLB 4-2 Core HL 1,3']}

Key Result

Proposition 1.1

For all $n\geq 0$, we have

Figures (8)

  • Figure 12: Three examples of 6-hooks
  • Figure 21: A tower
  • Figure 22: A 9-hook
  • Figure 23: The region of the hook
  • Figure 25: A $2t$-hook
  • ...and 3 more figures

Theorems & Definitions (19)

  • Proposition 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Remark
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 9 more