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The number of flats of a hyperplane arrangement is considered as ageneralization of the Bell number and the Stirling number of the second kind.Robert Gill gave the exponential generating function of the number of flats ofthe extended Catalan arrangements, using species. In this article, we introducethe species of flats of the extended Catalan and Shi arrangements and they aregiven by iterated substitution of species of sets and lists. Moreover, weenumerate the flats of these arrangements in terms of infinite matrices.

Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species