Square $\boldsymbol {q,t}$-lattice paths and $\boldsymbol {\nabla (p_n)}$

Abstract

The combinatorial q,t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q,t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n'th q,t-Catalan number is the for the module of diagonal harmonic alternants in 2n variables; it is also the coefficient of s 1 n in the Schur expansion of ∇(e n ). Using q,t-analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of ∇(e n ) and the of the diagonal harmonics modules. This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several q,t-analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the q,t-Catalan polynomials. We also conjecture.an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of ∇(p n ), the Hilbert series (∇(p n ), h 1 n), and the sign character (∇(p n ),s 1 n).