Generalized $q,t$-Catalan numbers

TL;DR

This paper introduces a combinatorial approach to generalized $q,t$-Catalan numbers, demonstrating their nonnegative integer coefficients and providing explicit combinatorial interpretations for sequences of length up to 4.

Contribution

It offers a new combinatorial perspective on generalized $q,t$-Catalan numbers, including explicit enumeration and symmetric chain decompositions for small cases.

Findings

Coefficients are nonnegative integers in many cases.

For sequences of length ≤ 4, coefficients count subdiagrams in Young diagrams.

Provides explicit symmetric chain decompositions for these diagrams.

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.