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