{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XD4B7VKP4KOCMC5WVQ5NHR33IC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4519cb043bd5f28d55932d2aab76e5e967fe977fc67773df8bbad24387b10cce","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-05T11:50:31Z","title_canon_sha256":"2ea14076ab5fa844a507d67fd38ee2da494f5ae5139f05d63900baec69ff76ce"},"schema_version":"1.0","source":{"id":"1612.01320","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.01320","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"arxiv_version","alias_value":"1612.01320v1","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01320","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"pith_short_12","alias_value":"XD4B7VKP4KOC","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XD4B7VKP4KOCMC5W","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XD4B7VKP","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:9fcfdb1089c704bf9cfe3940707e78c37a70bb5115985e0467f5a900d287ec33","target":"graph","created_at":"2026-05-18T00:11:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces correspondin","authors_text":"Deniz Kus, G. Arunkumar, R. Venkatesh","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-05T11:50:31Z","title":"Root multiplicities for Borcherds algebras and graph coloring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01320","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c8399ab3b8e434b129a8047ecbb705588c5f5885fa07a3bce93571bdd22f29ab","target":"record","created_at":"2026-05-18T00:11:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4519cb043bd5f28d55932d2aab76e5e967fe977fc67773df8bbad24387b10cce","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-05T11:50:31Z","title_canon_sha256":"2ea14076ab5fa844a507d67fd38ee2da494f5ae5139f05d63900baec69ff76ce"},"schema_version":"1.0","source":{"id":"1612.01320","kind":"arxiv","version":1}},"canonical_sha256":"b8f81fd54fe29c260bb6ac3ad3c77b408de1a683d1fa82994e7cb57e2809a409","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8f81fd54fe29c260bb6ac3ad3c77b408de1a683d1fa82994e7cb57e2809a409","first_computed_at":"2026-05-18T00:11:11.829439Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:11.829439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m6jDF+a1qgapFqyRZ+YBoiDK5jpP45Qs9qLUzsP7q5dea97ouKrysa8dwVwj2VUeWaj8XE7u4USSeXon05EKCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:11.830258Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.01320","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8399ab3b8e434b129a8047ecbb705588c5f5885fa07a3bce93571bdd22f29ab","sha256:9fcfdb1089c704bf9cfe3940707e78c37a70bb5115985e0467f5a900d287ec33"],"state_sha256":"c07cf854810d58e289d00b2f377747f3b64d3a892be82491e871dc461fe0799e"}