Representation Stability for Disks in a Strip

We consider the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$ exhibits a notion of first-order representation stability similar to Church--Ellenberg--Farb and Miller--Wilson's first-order representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large $w$ this disk configuration space exhibits notions of second- (and higher) order representation stability.