A proof of the q, t-Catalan positivity conjecture

Abstract

We present 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. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C n ( q, t) is the Hilbert series of the diagonal harmonic alternants in the variables ( x 1, x 2,…, x n ; y 1, y 2,…, y n ). Since 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( π), b( π) on Dyck paths π in the n × n square yielding C n ( q, t)=∑ π t a( π) q b( π) . Our proof is based on a recursion for C n ( q, t) suggested by a pair of statistics a( π), b( π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).