  We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and
give infinite families of congruences modulo $3$ (in arithmetic progression)
for $b_6(n)$. We also consider the number of the partitions of $n$ into
distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate
connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial
interpretations for these partition functions. In this context, we discover new
infinite families of linear inequalities involving Euler's partition function
$p(n)$. Infinite families of linear inequalities involving the $6$-regular
partition function $b_6(n)$ and the distinct partition function $Q_2(n)$ are
proposed as open problems.
