Recent work of the first author, Negut , and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov-Rozansky knot homology produces a family of polynomials in q and t labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The q, t-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients.For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for (4, n) rational q, t-Catalan numbers.
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 nonsymmetric Macdonald polynomials. Proposition 1.1. If m and n are coprime then m-stable affine permutations label the alcoves in a certain simplex D m n which is isometric to the m-dilated fundamental alcove. In particular, the number of m-stable affine permutations equals m n−1 .The simplex D m n (first defined in [10,27]) plays the central role in our study. We show that the alcoves in it naturally label various algebraic and geometric objects such as cells in certain affine Springer fibres and nonsymmetric Macdonald polynomials at q m = t n . We provide a clear combinatorial dictionary that allows one to pass from one description to another.We define two maps A, PS between the m-stable affine permutations and m n-parking functions and prove the following results about them. Theorem 1.2. Maps A and PS satisfy the following properties:(1) The map A is a bijection for all m and n.(2) The map PS is a bijection for m = kn ± 1. For m = kn + 1, it is equivalent to the Pak-Stanley labeling of Shi regions. 1 2 EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI (3) The map PS ○ A −1 generalizes the bijection ζ constructed by Haglund in [18]. More concretely, if one takes m = n + 1 and restricts the maps A and PS to minimal length right coset representatives of S n S n , then PS ○ A −1 specializes to Haglund's ζ. Remark 1.3. For m = n + 1 the bijection A is similar to the Athanasiadis-Linusson [5] labeling of Shi regions, but actually differs from it. Conjecture 1.4. The map PS is bijective for all relatively prime m and n.The map PS has an important geometric meaning. In [24] Lusztig and Smelt considered a certain Springer fibre F m n in the affine flag variety and proved that it can be paved by m n−1 affine cells. In [14,15] a related subvariety of the affine Grassmannian has been studied under the name of Jacobi factor, and a bijection between its cells and the Dyck paths in m × n rectangle has been constructed. In [20] Hikita generalized this combinatorial analysis and constructed a bijection between the cells in the affine Springer fiber and m n-parking functions (in slightly different terminology). He gave a quite involved combinatorial formula for the dimension of ...
ABSTRACT. Back in the nineties Pak and Stanley introduced a labeling of the regions of a kShi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph G. They introduced the G-Shi arrangement and a labeling of its regions by G-parking functions. They conjectured that their labeling is surjective, i.e. that every G-parking function appears as a label of a region of the G-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the k = 1 case. We generalize Hopkins-Perkinson's construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary k.
We study the relationship between rational slope Dyck paths and invariant subsets 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. Y n,m M n,m Y n,m Core n,m D G ζ A FIGURE 1. Rational Catalan maps in the coprime case
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