157 research outputs found
A Combinatorial Formula for Macdonald Polynomials
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.Comment: 29 page
Rational Parking Functions and LLT Polynomials
We prove that the combinatorial side of the "Rational Shuffle Conjecture"
provides a Schur-positive symmetric polynomial. Furthermore, we prove that the
contribution of a given rational Dyck path can be computed as a certain skew
LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel
and Ulyanov. The corresponding skew diagram is described explicitly in terms of
a certain (m,n)-core.Comment: 14 pages, 8 figure
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