Polynomial entropy on the $n$-fold symmetric product and its suspension

We prove that the polynomial entropy of the induced map $F_n(f)$ on the $n$-fold symmetric product of a compact space $X$ and its suspension are both equal to $n{h}_{pol}(f)$, when $f:X\to X$ is a homeomorphism with a finite chain recurrent set $\mathcal{C}\mathcal{R}(f)$. We also give a lower bound for the polynomial entropy on the suspension, for a homeomorphism $f$ with at least one wandering point, under certain assumptions.