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

## TL;DR

This paper introduces species-based methods to enumerate flats of extended Catalan and Shi arrangements, generalizing classical combinatorial numbers and providing explicit enumeration formulas.

## Contribution

It develops species of flats for these arrangements and derives enumeration formulas using iterated substitution and infinite matrices.

## Key findings

- Enumeration formulas for flats of extended Catalan arrangements
- Enumeration formulas for flats of Shi arrangements
- Representation of flats via infinite matrices

## 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.