Lagrange Inversion and Schur Functions
Journal of Algebraic Combinatorics: An International Journal
A proof of the q, t-Catalan positivity conjecture
Discrete Mathematics
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
The correlation functions of vertex operators and Macdonald polynomials
Journal of Algebraic Combinatorics: An International Journal
A proof of the q, t-square conjecture
Journal of Combinatorial Theory Series A
q,t-Fuβ---Catalan numbers for finite reflection groups
Journal of Algebraic Combinatorics: An International Journal
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
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 introduce a rational function Cn(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number {1 \over n+1} {2n \choose n}. We give supporting evidence by computing the specializations D_n(q) = C_n(q, 1/q)q^{n\choose 2} and Cn(q) = Cn(q, 1) = Cn(1,q). We show that, in fact, Dn(q) q -counts Dyck words by the major index and Cn(q) q -counts Dyck paths by area. We also show that Cn(q, t) is the coefficient of the elementary symmetric function en in a symmetric polynomial DHn(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that Cn(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DHn(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {Pμ(x; q, t)}μ which are best dealt with in λ-ring notation. In particular we derive here the λ-ring version of several symmetric function identities.