The classes of the quasihomogeneous Hilbert schemes of points on the plane

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.