The Combinatorics of Avalanche Dynamics

2011/11/22 by Manfred Denker, Denker, Manfred, Ana Rodrigues +1
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1111.5071

openalex publication_date 2011/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a simple and elementary proof of the identity ∑_r=1ⁿ∑_k₁,...,k_r≥ 1: ∑_i=1^r k_i= n \frac n! k₁!k₂!...k_r!k₁^k₂...k_r-1^k_r=(n+1)^n-1 where n∈ \mathbb N. A first application of this formula shows Cayley's theorem \citeCaley 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. \citeEHE and Levina \citeLevina is closely related to this dynamical representation.

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