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 Hilbert series for the module of diagonal harmonic alternants in $2n$ variables; it is also the coefficient of $s_{1^n}$ in the Schur expansion of $\nabla (e_n)$. Using $q,t$-analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of $\nabla (e_n)$ and the Hilbert series 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 $\nabla (p_n)$, the “Hilbert series” $\langle \nabla (p_n),h_{1^n}\rangle$, and the sign character $\langle \nabla (p_n),s_{1^n}\rangle$.