  Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the
property that corresponding to every prime $p$ there exists an integer $\ell$
such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed
results between the number of partitions of an integer $n$ whose parts are
taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ and the number of parts
of those partitions which are in a certain arithmetic progression.
