Note on partitions into polynomials with number of parts in an arithmetic progression

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