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 . We give supporting evidence by computing the specializations 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 _nin a symmetric polynomial DH_n(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 _n(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH_n(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 (x; q, t)} which are best dealt with in -ring notation. In particular we derive here the -ring version of several symmetric function identities.Work carried out under NSF grant support.