A Schröder generalization of Haglund's statistic on Catalan paths (Q1871380)

Summary: \textit{A. M. Garsia} and \textit{M. Haiman} [J. Algebr. Comb. 5, 191-244 (1996; Zbl 0853.05008)] 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. \textit{J. Haglund} later discovered [Adv. Math., in press], and with Garsia proved [Proc. Natl. Acad. Sci. USA 98, 4313-4316 (2001; Zbl 1066.05144)] the refined conjecture \(C_n(q,t) = \sum q^{\text{area}}t^{\text{bounce}}\). Here the sum is over all Catalan lattice paths and area and bounce have simple descriptions in terms of the path. In this article we give an extension of (area, bounce) to Schröder lattice paths, and introduce polynomials defined by summing \(q^{\text{area}}t^{\text{bounce}}\) over certain sets of Schröder paths. We derive recurrences and special values for these polynomials, and conjecture they are symmetric in \(q,t\). We also describe a much stronger conjecture involving rational functions in \(q,t\) and the \(\nabla\) operator from the theory of Macdonald symmetric functions.