A Remarkable q, t-Catalan Sequence and q-Lagrange Inversion

AbstractWe 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 $$\frac{1}{{n + 1}}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)$$ . We give supporting evidence by computing the specializations $$D_n \left( q \right) = C_n \left( {q{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \right)q^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)} $$ 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.