Verifying that the symmetric group $\mathsf{S}_m$ acts on $\mathbb{N}^m$
Show that $\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$