Chip-Firing on the Complete Split Graph: Motzkin Words and Tiered Parking Functions

We highlight some results from studying chip-firing on the the complete split graph [5]. In this work it is shown that recurrent states can be characterised in terms of Motzkin words and can also be characterised in terms of a new type of parking function that we call a tiered parking function. These new parking functions arise by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them.