A Schröder Generalization of Haglund's Statistic on Catalan Paths

Abstract

Garsia and Haiman (J. Algebraic. Combin. 5 (1996), 191-244) conjectured that a certain sum C_n(q, t) of rational functions in q, t reduces to a polynomial in q, t with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. 98 (2001), 4313-4316) the refined conjecture C_n(q, t) = . Here the sum is over all Catalan lattice paths and area and bounce have simple descriptions in terms of..