A positivity result in the theory of Macdonald polynomials

Abstract

We outline here a proof that a certain rational function C(n)(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 C(n)(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 (n) of Dyck paths Pi of length 2n yielding C(n)(q, t) = summation operator(pi) t(a(Pi))q(b(Pi)). Our proof is based on a recursion for C(n)(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.