This project is aimed to the study of cyclic sieving phenomenon (CSP) in certain combinatorial objects, and a generalization of the statistics of merging blocks and successions of set partitions.

There are two different approaches to proving instances of the cyclic sieving phenomenon. The first is combinatorial. This means that by explicitly evaluating the CSP polynomials at roots of unity and also counting the number of fixed points of the sets under the group actions. It may also involve the use of equivariant bijections to derive new CSP triples from the previously known ones. The second is using a representation theory, i.e., using vector spaces and diagonalization.

There is a proof that the set of standard young tableaux of a certain shape exhibits the CSP using the group action by promotion. The proof uses something called webs for 3-row case. This looks to be refined to cyclic descents (cdes). So, I would like to work on the refinement of the 3-row case to cdes. Also, since finding the analog of webs for the 4-row case is an open problem, I would like to find this analog.

The cyclic sieving phenomenon on the set of circular Dyck paths of height n and width w under the action of the generator which acts by cyclically shifting the area sequence one step looks to be refined by taking the number of circular peaks of the circular Dyck path into account. Thus, I would like to work on finding this refinement.

On the other hand, the joint distribution of the statistics of merging blocks and successions has been studied in the set of set partitions, and there is an involution that interchanges them for any set partition. I would, then, like to extend the study of these statistics into the set of type B partitions over [±n]. Also, I would like to study the 2-adic valuation of the Bell numbers of type B.