Some congruences for $\ell$-regular partitions with certain restrictions

Let ${\mathrm{p}\mathrm{o}\mathrm{d}}_{\ell }(n)$ and ${\mathrm{p}\mathrm{e}\mathrm{d}}_{\ell }(n)$ denote the number of $\ell$-regular partitions of a positive integer $n$ into distinct odd parts and the number of $\ell$-regular partitions of a positive integer $n$ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for ${\mathrm{p}\mathrm{o}\mathrm{d}}_{\ell }(n)$. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for ${\mathrm{p}\mathrm{o}\mathrm{d}}_5(n)$. We also proved a congruence relation for ${\mathrm{p}\mathrm{e}\mathrm{d}}_{\ell }(n)$. In particular, we established a congruence relation modulo 2 connecting ${\mathrm{p}\mathrm{o}\mathrm{d}}_{\ell }(n)$ and ${\mathrm{p}\mathrm{e}\mathrm{d}}_{\ell }(n)$.