A Combinatorial Formula for Macdonald Polynomials

Abstract

We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t) which had been conjectured by the first author. Corollaries to our main theorem include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger 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 K_{lambda,mu}(q,t) in the case that mu is a partition with parts less than or equal to 2.

29 pages

References in corpus (3)

• A recursion and a combinatorial formula for Jack polynomials
• Difference equations and symmetric polynomials defined by their zeros
• Integrality of two variable Kostka functions

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• Haglund's conjecture on 3-column Macdonald polynomials
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• Dual equivalence graphs II: Transformations on locally Schur positive graphs
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• Nested Hilbert schemes and the nested q,t-Catalan series
• Affine dual equivalence and k-Schur functions
• Paths and Kostka--Macdonald Polynomials
• Macdonald symmetry at $q=1$ and a new class of inv-preserving bijections on words
• A k-tableau characterization of k-Schur functions
• The Combinatorics of the Garsia-Haiman Modules for Hook Shapes