In this dissertation, we introduce combinatorial interpretations for three types of HallLittlewood polynomials (denoted Rλ, Pλ, and Qλ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments. / Ph. D.

The Kostka-Foulkes polynomials describe the change of basis between Schur polynomials and Hall-Littlewood polynomials. In this paper, we extend this idea to the family of BCn Macdonald spherical functions, with multiparameter Kostka-Foulkes polynomials acting as the change of basis from the BC_n spherical functions to the type Cn Schur polynomials. We develop a Kato-Lusztig formula that describes the multiparameter BCn-Kostka-Foulkes polynomials. / Master of Science / The work done in [11] gives a formula to calculate Kostka-Foulkes polynomials that convert between two other forms of polynomials. However, this only applies in specific instances. In this paper, we generalize those ideas to allow for more parameters, and find that a similar formula still holds.