New Congruences for $2$-Color Partitions and $t$-Core Partitions

Résumé

Let $c_N(n)$ denote the number of $2$-color partition of $n$ subject to the restriction that one of the colors appears only in parts that are divisible by $N$. If $t$ is a positive integer, then a partition of a nonnegative integer $n$ is a $t$-core if none of the hook numbers of the associated Ferrers-Young diagram is a multiple of $t$. Let $a_t(n)$ denote the number of $t$-core partitions of $n$. In this paper, we obtain new congruences modulo $3$ for the $2$-color partition function $c_{11}(n)$, $t$-core partition functions $a_5(n)$ and $a_{11}(n)$.