The sandpile model on the complete split graph, Motzkin words, and tiered parking functions

TL;DR

This paper classifies recurrent states of the Abelian sandpile model on the complete split graph using combinatorial objects like Motzkin words, necklaces, and introduces tiered parking functions, providing enumeration formulas and spanning tree counts.

Contribution

It introduces a new characterization of recurrent states of the ASM on the complete split graph via Motzkin words, necklaces, and tiered parking functions, along with enumeration formulas.

Findings

Characterization of recurrent states using Motzkin words and necklaces.

Introduction of tiered parking functions for recurrent state characterization.

Enumeration formulas for recurrent configurations and spanning trees.

Abstract

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised 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. We also enumerate the different sets of recurrent configurations studied in this paper, and in doing so derive a formula for the number of spanning trees of the complete split graph that uses a bijective Pr\"ufer code argument.