Topology of the punctual Hilbert schemes of plane curve singularities with a single Puiseux pair

Piontkowski proved the existence of affine cell decompositions of Jacobian factors of plane curve singularities with a single Puiseux pair. He also provided a combinatorial description of the Euler numbers and Betti numbers of these Jacobian factors. Following his results, Oblomkov, Rasmussen, and Shende demonstrated the existence of affine cell decompositions of punctual Hilbert schemes for the same type of singularity. In the present paper, we revisit their theorem from a computational perspective and describe the Euler numbers and Betti numbers of the punctual Hilbert schemes.