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

Marko Thiel

Abstract


The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fuß-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.


Keywords


Fuss-Catalan combinatorics; Catalan arrangement; Floors; Ceilings

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