{"paper":{"title":"The Combinatorics of Avalanche Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ana Rodrigues, Manfred Denker","submitted_at":"2011-11-22T00:48:48Z","abstract_excerpt":"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 Eu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5071","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}