Compactified Jacobians and q,t-Catalan numbers, II

Abstract

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.

Compositional (km,kn)-Shuffle Conjectures

“…This is a reformulation of Hikita's definition of the dinv of an (m, n)-parking function first introduced by Gorsky and Mazin in [13].…”

Compactified Jacobians and q,t-Catalan numbers, I

“…The map G coincides with the bijection from [14,15] transforming a pair of statistics (area, dinv) into (bounce, area) statistics. We further explore this connection in our next paper [11].…”

“…If (p, q) = (n, n + 1), then dim D = n 2 − dinv(D). Therefore, = g(8) = 1, and g (11) = g (14) = 0. Therefore, dim = 7.…”

“…Finally, in Section 5 we mention a possible relation of our results to HOMFLY homology of torus knots. We continue our study of combinatorics of Jacobi factors in [11].…”

Combinatorics of the zeta map on rational Dyck paths

“…Many details about the zeta map have been gathered and unified in a comprehensive article by Armstrong, Loehr, and Warrington [3], including progress on proving its bijectivity in certain cases such as (a, am ± 1)-Dyck paths [12,19] (which is associated to the Fuss-Catalan numbers). The zeta map was shown to be a bijection in these special cases by way of a "bounce path" by which zeta inverse could be computed.…”