**Abstract** The Hilbert scheme $\text{Hilb}^n X$ of $n$ points on a quasi-projective variety $X$ is a geometrical object introduced by Grothendieck and it has a prominent rôle in many areas of algebraic geometry. Recently, many variants of $\text{Hilb}^n X$ have been introduced. My talk will focus on the double nested Hilbert scheme of points on $X$ defined by S. Monavari. Specifically, I will explain how, when $X$ is a smooth irreducible curve, its geometry is influenced by the combinatorics of reverse plane partitions and exhibits several pathologies.

This is a joint project with Lella, Monavari, Ricolfi, Sammartano.