{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:XUJQTSRPCHP7KU626XITCVVEAY","short_pith_number":"pith:XUJQTSRP","schema_version":"1.0","canonical_sha256":"bd1309ca2f11dff553daf5d13156a4061b0a3c1292fa75ee1f91c091c664ce32","source":{"kind":"arxiv","id":"1907.08864","version":1},"attestation_state":"computed","paper":{"title":"Generalized chromatic polynomials of graphs from Heaps of pieces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G Arunkumar","submitted_at":"2019-07-20T20:05:43Z","abstract_excerpt":"Let $G$ be a simple graph and let $\\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a "},"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":"1907.08864","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-20T20:05:43Z","cross_cats_sorted":[],"title_canon_sha256":"3083ffc544d431af14edd12bd5d743399499056b43532bd42582318c59c0707e","abstract_canon_sha256":"846fbb58ba6086a2dfa5e0357826504174c1a1d6cf888e637d4e636855fb0046"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:02.959785Z","signature_b64":"0TQZ8FFEXtwpw72z7S2jacEukopNKuw+EZWFr3VqbyGKO+qcG2U8SovenaUeBC03ORZXnmD4Qf8M9GIvgX2WAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd1309ca2f11dff553daf5d13156a4061b0a3c1292fa75ee1f91c091c664ce32","last_reissued_at":"2026-05-17T23:40:02.959210Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:02.959210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized chromatic polynomials of graphs from Heaps of pieces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G Arunkumar","submitted_at":"2019-07-20T20:05:43Z","abstract_excerpt":"Let $G$ be a simple graph and let $\\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08864","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":"1907.08864","created_at":"2026-05-17T23:40:02.959288+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.08864v1","created_at":"2026-05-17T23:40:02.959288+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08864","created_at":"2026-05-17T23:40:02.959288+00:00"},{"alias_kind":"pith_short_12","alias_value":"XUJQTSRPCHP7","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"XUJQTSRPCHP7KU62","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"XUJQTSRP","created_at":"2026-05-18T12:33:33.725879+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/XUJQTSRPCHP7KU626XITCVVEAY","json":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY.json","graph_json":"https://pith.science/api/pith-number/XUJQTSRPCHP7KU626XITCVVEAY/graph.json","events_json":"https://pith.science/api/pith-number/XUJQTSRPCHP7KU626XITCVVEAY/events.json","paper":"https://pith.science/paper/XUJQTSRP"},"agent_actions":{"view_html":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY","download_json":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY.json","view_paper":"https://pith.science/paper/XUJQTSRP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.08864&json=true","fetch_graph":"https://pith.science/api/pith-number/XUJQTSRPCHP7KU626XITCVVEAY/graph.json","fetch_events":"https://pith.science/api/pith-number/XUJQTSRPCHP7KU626XITCVVEAY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY/action/storage_attestation","attest_author":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY/action/author_attestation","sign_citation":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY/action/citation_signature","submit_replication":"https://pith.science/pith/XUJQTSRPCHP7KU626XITCVVEAY/action/replication_record"}},"created_at":"2026-05-17T23:40:02.959288+00:00","updated_at":"2026-05-17T23:40:02.959288+00:00"}