A combinatorial formula for Macdonald polynomials

Abstract

We prove a combinatorial formula for the Macdonald polynomialH~μ(x;q,t)\tilde {H}_{\mu }(x;q,t)which had been conjectured by Haglund. Corollaries to our main theorem include the expansion ofH~μ(x;q,t)\tilde {H}_{\mu }(x;q,t)in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi’s combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficientsK~λμ(q,t)\tilde {K}_{\lambda \mu }(q,t)in the case thatμ\muis a partition with parts≤2\leq 2.