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 $(SSYT(\lambda ,\mu ),⟨{\mathrm{\partial }}^{b+2}⟩,f(q))$ exhibits the cyclic sieving phenomenon, where $\mathrm{\partial }$ is the jeu de taquin promotion operator and $f(q)$ is a modified Kostka-Foulkes polynomial ${\tilde{K}}_{\lambda ,\mu }(q)$, up to a power of $q$.