The Combinatorics of Avalanche Dynamics

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.