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Proof of Singh and Barman's conjecture on hook length biases

Hongshu Lin, Wenston J. T. Zang

TL;DR

The paper proves Singh and Barman's conjecture on hook-length biases by establishing $b_{t+1,2}(n) \geq b_{t,2}(n)$ for $t \geq 3$, $n \geq 0$ using a generating-function framework. It decomposes the difference into six components and interprets each via combinatorial constructions, including $\text{OPO}$-overpartitions, then proves nonnegativity through explicit injections in both odd and even $t$ cases (with a small adjustment at $t=4$). This yields a complete, bijective-style proof of the conjecture and extends the toolkit for analyzing hook-length biases in $t$-regular partitions. The results have implications for the combinatorial understanding of hook-length distributions and their relationships across regular partition families.

Abstract

Let $b_{t,i}(n)$ denote the total number of $i$-hooks in $t$-regular partitions of $n$. Singh and Barman conjectured that $b_{t+1,2}(n) \geq b_{t,2}(n)$ holds for all $t\ge 3$ and $n\ge 0$. This conjecture was known to hold for $t=3$ due to work of Barman Mahanta and Singh. In this paper, we prove this conjecture.

Proof of Singh and Barman's conjecture on hook length biases

TL;DR

The paper proves Singh and Barman's conjecture on hook-length biases by establishing for , using a generating-function framework. It decomposes the difference into six components and interprets each via combinatorial constructions, including -overpartitions, then proves nonnegativity through explicit injections in both odd and even cases (with a small adjustment at ). This yields a complete, bijective-style proof of the conjecture and extends the toolkit for analyzing hook-length biases in -regular partitions. The results have implications for the combinatorial understanding of hook-length distributions and their relationships across regular partition families.

Abstract

Let denote the total number of -hooks in -regular partitions of . Singh and Barman conjectured that holds for all and . This conjecture was known to hold for due to work of Barman Mahanta and Singh. In this paper, we prove this conjecture.
Paper Structure (3 sections, 21 theorems, 62 equations, 2 figures, 4 tables)

This paper contains 3 sections, 21 theorems, 62 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm-com-int-umn']} for odd number $t$
  3. A proof of Theorem \ref{['thm-com-int-umn']} for even number $t$

Key Result

Theorem 1.1

For all integers $n > 3$, we have $b_{3,2}(n) \geq b_{2,2}(n)$.

Figures (2)

  • Figure 1.1: The Young diagram of partition $(6,4,4,3,2,1,1)$
  • Figure 1.2: The hook lengths of $(6,4,4,3,2,1,1)$

Theorems & Definitions (22)

  • Theorem 1.1: Singh-Barman-2025
  • Theorem 1.1: Singh-Barman-2025
  • Conjecture 1.2: Singh-Barman-2025
  • Theorem 1.3: Barman-2025
  • Theorem 1.4: Kim
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Kim
  • Lemma 2.2
  • Lemma 2.3
  • ...and 12 more