The Combinatorics of Avalanche Dynamics

TL;DR

This paper presents a simple proof of a combinatorial identity related to avalanche dynamics, connects it to Cayley's theorem on trees, and explores its implications for avalanche size distributions in dynamical systems.

Contribution

It provides an elementary proof of a key combinatorial identity and applies it to analyze avalanche size distributions in dynamical systems and related probabilistic models.

Findings

The identity is equivalent to Cayley's theorem on trees.

The formula describes the distribution of avalanche sizes.

Applications include general dynamical systems and urn models.

Abstract

We give a simple and elementary proof of the identity $$\sum_{r=1}^n\sum_{k_1,...,k_r\ge 1: \sum_{i=1}^r k_i= n} \frac {n!} {k_1!k_2!...k_r!}k_1^{k_2}...k_{r-1}^{k_r}=(n+1)^{n-1}$$ where $n\in \mathbb N$. A first application of this formula shows Cayley's theorem \cite{Caley} on the number of trees with $n+1$ vertices (in fact the formula is equivalent to Cayley's result). A second application gives the distribution of avalanche sizes, which can be deduced for general dynamical systems and also as a bilogically motivated urn model in probability. In particular, the law of avalanche sizes in Eurich et al. \cite{EHE} and Levina \cite{Levina} is closely related to this dynamical representation.