Central-firing of type A_{2n} with initial weight 0

In this thesis a variant of the chip-firing game introduced by Hopkins, McConville and Propp in [1], called the labeled chip-firing on Z, is studied. We will first explore the basic properties and examples of this game. We will then show, how one can see this game as a binary relation on the weight lattice of Type A root system. It is then a natural step to generalize it to other root systems, which was done by Galashin, Hopkins, McConville and Postnikov in [2] and [3]. After reviewing the basics of central-firing introduced in these papers, we examine the central-firing of type A_{2n} with initial weight 0 in Chapter 5. Finally, we study the restrictions in Lemma 12 of [1] in more detail, and conjecture that the number of permutations with maximum number of inversions allowed by this lemma is given by the Catalan numbers.