Conjectures on the Quotient Ring by Diagonal Invariants
Journal of Algebraic Combinatorics: An International Journal
Factorizations of Pieri rules for Macdonald polynomials
Proceedings of the 4th conference on Formal power series and algebraic combinatorics
A Remarkable q, t-Catalan Sequence and q-Lagrange Inversion
Journal of Algebraic Combinatorics: An International Journal
t, q-Catalan numbers and the Hilbert scheme
Discrete Mathematics - selected papers in honor of Adriano Garsia
Enumeration of ad-nilpotent b-ideals for simple Lie algebras
Advances in Applied Mathematics - Special issue: Memory of Rodica Simon
Permutation statistics and the q, t-Catalan sequence
European Journal of Combinatorics
A proof of the q, t-square conjecture
Journal of Combinatorial Theory Series A
A continuous family of partition statistics equidistributed with length
Journal of Combinatorial Theory Series A
Bases for diagonally alternating harmonic polynomials of low degree
Journal of Combinatorial Theory Series A
A computational and combinatorial exposé of plethystic calculus
Journal of Algebraic Combinatorics: An International Journal
Compactified Jacobians and q,t-Catalan numbers, I
Journal of Combinatorial Theory Series A
An explicit formula for ndinv, a new statistic for two-shuffle parking functions
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Compactified Jacobians and q,t-Catalan numbers, II
Journal of Algebraic Combinatorics: An International Journal
A three shuffle case of the compositional parking function conjecture
Journal of Combinatorial Theory Series A
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We present 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. 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 Cn(q, t) is the Hilbert series of the diagonal harmonic alternants in the variables (x1,x2,...,xn;y1,y2,...,yn). Since 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(π), b(π) on Dyck paths π in the n × n square yielding Cn(q, t) = Σπta(π)qb(π). Our proof is based on a recursion for Cn(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).