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arXiv · 1311.5718

On floors and ceilings of the k-Catalan arrangement

Marko Thiel
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Abstract

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.

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Marko Thiel. 2014-11-05. On floors and ceilings of the k-Catalan arrangement. https://arxiv.org/abs/1311.5718

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