{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DIXXRJ65TT6ETO3L2OHSECDRPA","short_pith_number":"pith:DIXXRJ65","schema_version":"1.0","canonical_sha256":"1a2f78a7dd9cfc49bb6bd38f220871780eb051360c0dbc613e5ce61f6936539f","source":{"kind":"arxiv","id":"1111.5071","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.5071","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-22T00:48:48Z","cross_cats_sorted":[],"title_canon_sha256":"b133748bee62ae4e201c58d360371cc3987040ff03f96bc0c134f9acc5cbe235","abstract_canon_sha256":"46ce06b9f15a86070d8145589126317833a0452562859dd9274500bf92b82f6d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:53.516006Z","signature_b64":"0KEiFrb5ZiWm35iDPpnIAWx4t5NuZrVaxMjc7jQaoO6ig4nmhwweH/eUHaFzS90iz7eiIS7KYz+wWEkuvUzzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a2f78a7dd9cfc49bb6bd38f220871780eb051360c0dbc613e5ce61f6936539f","last_reissued_at":"2026-05-18T04:07:53.515493Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:53.515493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1111.5071","created_at":"2026-05-18T04:07:53.515567+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.5071v1","created_at":"2026-05-18T04:07:53.515567+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5071","created_at":"2026-05-18T04:07:53.515567+00:00"},{"alias_kind":"pith_short_12","alias_value":"DIXXRJ65TT6E","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DIXXRJ65TT6ETO3L","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DIXXRJ65","created_at":"2026-05-18T12:26:26.731475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA","json":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA.json","graph_json":"https://pith.science/api/pith-number/DIXXRJ65TT6ETO3L2OHSECDRPA/graph.json","events_json":"https://pith.science/api/pith-number/DIXXRJ65TT6ETO3L2OHSECDRPA/events.json","paper":"https://pith.science/paper/DIXXRJ65"},"agent_actions":{"view_html":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA","download_json":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA.json","view_paper":"https://pith.science/paper/DIXXRJ65","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.5071&json=true","fetch_graph":"https://pith.science/api/pith-number/DIXXRJ65TT6ETO3L2OHSECDRPA/graph.json","fetch_events":"https://pith.science/api/pith-number/DIXXRJ65TT6ETO3L2OHSECDRPA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA/action/storage_attestation","attest_author":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA/action/author_attestation","sign_citation":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA/action/citation_signature","submit_replication":"https://pith.science/pith/DIXXRJ65TT6ETO3L2OHSECDRPA/action/replication_record"}},"created_at":"2026-05-18T04:07:53.515567+00:00","updated_at":"2026-05-18T04:07:53.515567+00:00"}