The goal of this talk is to algebraically model the S_r-equivariant homotopy type of the configuration space of r labeled and distinct points in d-dimensional Euclidean space. We will present and compare two models: the Barratt-Eccles simplicial set and the multisimplicial set of 'surjections'. Moreover, we will introduce multisimplicial sets and discuss their connection to more well-known simplicial sets. Multisimplicial sets can model homotopy types using fewer cells, making them a highly useful tool. Following this, we will explore in detail how to recognize configuration spaces in the aforementioned models by playing with a graph poset. An explicit relationship between the models will also be presented. Joint work with Anibal M. Medina-Mardones and Paolo Salvatore.