  The set of dominant regions of the $k$-Catalan arrangement of a
crystallographic root system $\Phi$ is a well-studied object enumerated by the
Fu{\ss}-Catalan number $Cat^{(k)}(\Phi)$. 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.
