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Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the numberof $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmeticprogressions $r_1 \bmod{m_1}$ 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 combinatorialproof of this result using abaci and the theory of $t$-cores and $t$-quotients.

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