Every modularly extended matroid has a divisional flag, so any hyperplane arrangement whose dependence matroid is modularly extended is divisionally free.
Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species
1 Pith paper cite this work. Polarity classification is still indexing.
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Pith paper citing it
abstract
The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices.
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Modular Construction of Free Hyperplane Arrangements
Every modularly extended matroid has a divisional flag, so any hyperplane arrangement whose dependence matroid is modularly extended is divisionally free.