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

Abstract

Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a\mathrm{mod}b$. For $t\in \{2,3\}$, arithmetic progressions $r_1\mathrm{mod}{m}_1$ and $r_2\mathrm{mod}{m}_2$ 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.

7 pages