Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.2

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

Norihiro Nakashima
Department of Mathematics
Nagoya Institute of Technology
Nagoya, Aichi 466-8555

Shuhei Tsujie
Department of Mathematics
Hokkaido University of Education
Asahikawa, Hokkaido 070-8621


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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(Concerned with sequences A000012 A000110 A000142 A000262 A000670 A002866 A008277 A025168 A034001 A034177 A034325 A048786 A050351 A050352 A050353 A075729 A079621 A088729 A105278 A109092 A308281 A308282 A308440 A321837 A321847 A321848.)

Received June 18 2021; revised version received September 29 2021. Published in Journal of Integer Sequences, September 30 2021.

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