  The $q,t$-Catalan numbers can be defined using rational functions, geometry
related to Hilbert schemes, symmetric functions, representation theory, Dyck
paths, partition statistics, or Dyck words. After decades of intensive study,
it was eventually proved that all these definitions are equivalent. In this
paper, we study the similar situation for higher $q,t$-Catalan numbers, where
the equivalence of the algebraic and combinatorial definitions is still
conjectural. We compute the limits of several versions of the modified higher
$q,t$-Catalan numbers and show that these limits equal the generating function
for integer partitions. We also identify certain coefficients of the higher
$q,t$-Catalan numbers as enumerating suitable integer partitions, and we make
some conjectures on the homological significance of the Bergeron-Garsia nabla
operator.
