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The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions

Hongshu Lin, Wenston J. T. Zang

Abstract

Let $b_{n,k}$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh and Barman raised the question of finding the relation between $b_{t,2}(n)$ and $b_{t,1}(n)$. Kim showed that there exists $N$ such that $b_{t,2}(n)\ge b_{t,1}(n)$ and $b_{t,2}(n) \geq b_{t,3}(n)$ for $n>N$. In this paper, we find an explicit bound of $N=O(t^5)$ for $b_{t,2}(n)\geq b_{t,1}(n)$ and show that $b_{t,2}(n) \geq b_{t,3}(n)$ for all $n\ge 4$.

The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions

Abstract

Let denote the number of hooks of length in all the -regular partitions of . Singh and Barman raised the question of finding the relation between and . Kim showed that there exists such that and for . In this paper, we find an explicit bound of for and show that for all .
Paper Structure (3 sections, 19 theorems, 78 equations, 2 figures)

This paper contains 3 sections, 19 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A proof of Theorem \ref{['thm-com-int-umn']}
  3. A proof of Theorem \ref{['thm-bt2-bt3']}

Key Result

Theorem 1.1

Let $t\geq 2$ be an integer. For sufficiently large integers $n$,

Figures (2)

  • Figure 1.1: The Young diagram of partition $(5,3,3,2,1,1)$
  • Figure 1.2: The hook lengths of $(5,3,3,2,1,1)$

Theorems & Definitions (19)

  • Theorem 1.1: Kim
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Singh-Barman-2024
  • Theorem 2.2: Kim
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 9 more