We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion.
The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial.
The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer.
Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon.
One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.