Trapezoidal lattice paths and multivariate analogues

AbstractIn 1996, Garsia and Haiman introduced a bivariate analogue of the Catalan numbers that counts multiplicities of the sign character in a certain doubly graded Sn-module. Haglund conjectured a combinatorial interpretation for this sequence by defining two statistics on the set of lattice paths staying inside the triangle bounded by x=0, y=x, and y=n. This conjecture was eventually proved by Garsia and Haglund. Later, the present author introduced similar statistics for lattice paths staying inside other triangles, whose generating function is conjectured to give the higher q,t-Catalan sequences of Garsia and Haiman. This article generalizes these combinatorial statistics to lattice paths within certain trapezoidal regions. We introduce a five-variable generating function for these paths and prove formulas, bijections, and recursions involving this generating function. An open question is to find representation-theoretical interpretations for these statistics analogous to those previously conjectured for the paths staying inside triangular regions