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A combinatorial formula for Macdonald polynomials. (English) Zbl 1061.05101
Summary: We prove a combinatorial formula for the Macdonald polynomial $$\widetilde{H}_{\mu}(x;q,t)$$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of $$\widetilde{H}_{\mu}(x;q,t)$$ 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 coefficients $$\widetilde{K}_{\lambda \mu}(q,t)$$ in the case that $$\mu$$ is a partition with parts $$\leq 2$$.

##### MSC:
 05E10 Combinatorial aspects of representation theory 05A30 $$q$$-calculus and related topics
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