  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.
