Cyclic Sieving for a Family of Semistandard Tableaux

We give a new cyclic sieving phenomenon for semistandard Young tableaux $SSYT(\lambda,\mu)$ of shape $\lambda=(m,n^b)$ and content $\mu$, a $(b+2)$-tuple. We prove that $(SSYT(\lambda,\mu),\langle \partial^{b+2} \rangle, f(q))$ exhibits the cyclic sieving phenomenon, where $\partial$ is the jeu de taquin promotion operator and $f(q)$ is a modified Kostka-Foulkes polynomial $\widetilde{K}_{\lambda,\mu}(q)$, up to a power of $q$.