A positivity result in the theory of Macdonald polynomials
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Communicated by Ronald L. Graham, University of California at San Diego, La Jolla, CA (received for review October 1, 2000)
Abstract
We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection 𝒟n of Dyck paths Π of length 2n yielding Cn(q, t) = ∑π t a(Π) q b(Π). Our proof is based on a recursion for Cn(q, t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund's conjecture.
Footnotes
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↵ † To whom reprint requests should be addressed. E-mail: jhaglund{at}math.ucsd.edu.
- Copyright © 2001, The National Academy of Sciences
