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JOURNAL OF
ALGEBRAIC
COMBINATORICS
  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

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

A.M. Garsia and M. Haiman

DOI: 10.1023/A:1022476211638

Abstract

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 ) \frac{1}{{n + 1}}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right) . We give supporting evidence by computing the specializations D n ( q ) = C n ( q1 \mathord
 / \vphantom 1 q q ) q ( *20 c n 2 ) 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 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.

Pages: 191–244

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

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References

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