A new analogue of $t$-core partitions
Volume 199 / 2021
Abstract
By analogy with $t$-core partitions, we study $\overline {a}_t(n)$, given by $\sum _{n=0}^{\infty }\overline {a}_t(n)q^n\break ={\phi (-q^t)^t}/{\phi (-q)}.$ We obtain multiplicative formulas and arithmetic properties of $\overline {a}_{t}(n)$ for $t\in \{3,4,8\}$. Moreover, if $8n+5$ is square-free then we prove $\overline {a}_{4}(2^{2\alpha }n)=12h(-4n)$, where $\alpha $ is any positive integer and $h(D)$ denotes the class number of binary quadratic forms of discriminant $D$. For a fixed positive integer $j$ and prime numbers $p_i\geq 5,$ we also show that $\overline {a}_{t}(n)$ is almost always divisible by $p_i^j$ if $p_i^{2a_i}\geq t$, where $t=p_1^{a_1} \dots p_m^{a_m}.$ Additionally, using modular forms we prove a Ramanujan type congruence for $\overline {a}_{5}$ modulo $5$.
