On floors and ceilings of the k-Catalan arrangement

The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\mathrm{\Phi }$ is a well-studied object enumerated by the Fu{\ss}-Catalan number $Ca{t}^{(k)}(\mathrm{\Phi })$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.