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A Proof of the q,t-Catalan Positivity Conjecture

By A. M. Garsia and J. Haglund


. 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 coe#cients. This has been an open problem since 1994. The precise form of the conjecture is given in the J. Algebraic Combin. 5 (1996), no. 3, 191--244, 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(#),b(#)on Dyck paths # in the nn square yielding C n (q, t)= P # t a(#) q b(#) . Our proof is based on a recursion for C n (q, t) suggested by a pair of statistics a(#),b(#) recently proposed by J. 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 m..

Year: 2000
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