(based on joint work with ruofan jiang) the hilbert scheme of points on a variety  parametrizes -dimensional quotients of the structure sheaf. when  is a planar singular curve, its enumerative invariants have close relation to mathematical physics, knot theory and combinatorics. in this talk, we investigate two analogous moduli spaces, one being a direct generalization of the hilbert scheme. our results reveal their surprising relations to hall polynomials, matrix equations, modular forms, etc.