We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m, n)-core.

We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite-dimensional representations of DAHA and non-symmetric Macdonald polynomials.

We study the relationship between rational slope Dyck paths and invariant sub- sets of ℤ, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn,dm)-Dyck paths and d-tuples of (n,m)-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.

We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of N,M-torus links and the rational q,t-Catalan power series for general positive N,M.