  In this paper we give a formula for the classes (in the Grothendieck ring of
complex quasi-projective varieties) of irreducible components of
$(1,k)$-quasi-homogeneous Hilbert schemes of points on the plane. We find a new
simple geometric interpretation of the $q,t$-Catalan numbers. Finally, we
investigate a connection between $(1,k)$-quasi-homogeneous Hilbert schemes and
homogeneous nested Hilbert schemes.
