ON THE NUMBER OF 2-HOOKS AND 3-HOOKS OF INTEGER PARTITIONS

Abstract

AbstractLet $p_t(a,b;n)$ 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 [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma10 (2022), Article no. e49] 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.