Combinatorics of Macdonald polynomials and cyclic sieving

Sammanfattning: In this thesis, we study the non-symmetric Macdonald polynomials E_λ (x;q,t) at t=0 from a combinatorial point of view, using the combinatorial formula found by J. Haglund, M. Haiman, and N. Loehr. Our primary focus is when λ is a partition. We summarize the known theory about this specialization and prove some new results related to this combinatorial formula. We also define the cyclic sieving phenomenon (CSP). For rectangular λ, we present an instance of cyclic sieving with E_λ (1,q,q^2,...,q^(k-1);1,0) as CSP-polynomial. We also conjecture another instance of CSP with E_λ (1,1,...,1;q,0) as CSP-polynomial. This conjecture generalizes a previously known CSP-triple. Furthermore, we prove this conjecture in the case when is λ an m×2 diagram.