Congruences modulo $7$ and $11$ for certain two restricted partition functions

Abstract

For an integer $c\geq 1$, let $a_c(n)$ count the number of generalized cubic partitions of $n$, which are partitions of $n$ whose even parts may appear in $c$ different colors, and $d_c(n)$ count the number of partitions obtained by adding the links of the $c$-elongated plane partition diamonds of length $n$. We prove in this note infinite families of congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ by employing elementary $q$-series techniques. These results generalize particular congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ recently found by Dockery, and Baruah, Das, and Talukdar, respectively, using modular forms.