Classes of Weierstrass points on genus $2$ curves

We study the codimension $n$ locus of curves of genus $2$ with $n$ distinct marked Weierstrass points inside the moduli space of genus $2$, $n$-pointed curves, for $n \leq 6$. We give a recursive description of the classes of the closure of these loci inside the moduli space of stable curves. For $n\leq 4$, we express these classes using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves. Similarly, we express the closure of these classes inside the moduli space of curves of compact-type for all $n$. This is a first step in the study of the structure of hyperelliptic classes in all genera.