On the Number of 2-Hooks and 3-Hooks of Integer Partitions

TL;DR

This paper provides a direct combinatorial proof for the distribution of 2- and 3-hooks in integer partitions, complementing previous modular form-based results by employing abaci and t-core theory.

Contribution

It introduces a combinatorial proof for the vanishing of certain partition counts related to 2- and 3-hooks, expanding understanding beyond modular form approaches.

Findings

Established vanishing of partition counts on specific arithmetic progressions.

Connected hook counts to abaci and t-core theory for combinatorial analysis.

Provided a new proof technique complementing existing modular form methods.

Abstract

Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmetic progressions $r_1 \bmod{m_1}$ and $r_2 \bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males, and Ono using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of $t$-cores and $t$-quotients.