Abacus-tournament models for Hall–Littlewood polynomials

Abstract

In 2010, the first author introduced a combinatorial model for Schur polynomials based on labeled abaci. We generalize this construction to give analogous models for the Hall–Littlewood symmetric polynomials Pλ, Qλ, and Rλ using objects called abacus-tournaments. We introduce various cancellation mechanisms on abacus-tournaments to obtain simpler combinatorial formulas and explain why these polynomials are divisible by certain products of t-factorials. These tools are then applied to give bijective proofs of several identities involving Hall–Littlewood polynomials, including the Pieri rule that expands the product Pμek into a linear combination of Hall–Littlewood polynomials.