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

Garsia, A.M. ; Haiman, M.

Journal of Algebraic Combinatorics, Tome 5 (1996) no. 3, pp. 191-244.

Résumé

Summary: 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 $\frac1 n + 1( *20 c 2 n n )$ frac1n + 1$left( beginarray*20c 2n n endarray right)$ . We give supporting evidence by computing the specializations $D _ n ( q ) = C _ n ( q1 mathord / vphantom1 q q ) q ^( *20 c n 2 )$ D\_n $left( q \right) = C$_n $left( q1 mathordleft/ vphantom1 q right. kern-nulldelimiterspace q right)$q^{$left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)}$ 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 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 $P \_$ mgr( x; q, t) _ mgr which are best dealt with in $lambda$-ring notation. In particular we derive here the $lambda$-ring version of several symmetric function identities.

Détail

Keywords: Catalan number, diagonal harmonic, Macdonald polynomial, Lagrange inversion