A remarkable q,t-Catalan sequence and q-Lagrange inversion

We introduce a rational function C-n (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/n+1((2n)(n))). evidence by computing the specializations D-n(q) = C-n(q, 1/q)q((2n)) and C-n(q) = C-n(q, 1) = C-n(1, q). We show that, in fact, D-n (q) q-counts Dyck words by the major index and C-n (q) q-counts Dyck paths by area. We also show that C-n(q, t) is the coefficient of the elementary symmetric function e(n) 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 C,(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH, (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 (P-mu(x; q, t)}(mu) which are best dealt with in h-ring notation. In particular we derive here the A-ring version of several symmetric function identities.