On Floors and Ceilings of the $k$-Catalan Arrangement

The set of dominant regions of the k-Catalan arrangement of a crystallographic root system Φ is a well-studied object enumerated by the Fuß-Catalan number Cat(k)(Φ). 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.