{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:XTUBC5D7TW3EWVGOQI3RZQFDCM","short_pith_number":"pith:XTUBC5D7","schema_version":"1.0","canonical_sha256":"bce811747f9db64b54ce82371cc0a31318c8bdd917dd21dcd7f1fbefb9ffbf0e","source":{"kind":"arxiv","id":"2509.02841","version":3},"attestation_state":"computed","paper":{"title":"Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Isaiah Siegl","submitted_at":"2025-09-02T21:20:53Z","abstract_excerpt":"Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are $e$-positive. Finding a combinatorial interpretation for these $e$-coefficients remains a major open problem. One approach is to look for combinatorial interpretations which are subsets of Gasharov's $P$-tableaux. Towards this goal, we introduce sets of strong and powerful $P$-tableaux, and use them to find combinatorial interpretations for various $e$-coefficients of the chromatic symmetric function $X_{inc(P)}(\\mathbf{x}, "},"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":"2509.02841","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-09-02T21:20:53Z","cross_cats_sorted":[],"title_canon_sha256":"b504e638fc6e531c16e9566c9b188c24c45f22008b8076ce303d7ce970029d1d","abstract_canon_sha256":"68f899b6de59f3be86d5dc9adbdaae7c3dd4b66d07025c8382ca8e8433f1e689"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T00:30:17.561381Z","signature_b64":"ou82TjcsXwBf2OWp8mMZV9kalXcVQpG9JLNUQyMO2Sv5FqQL3LiXMJUwg/SWEYuQtCekYkNeYNc5AhK+Q2kmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bce811747f9db64b54ce82371cc0a31318c8bdd917dd21dcd7f1fbefb9ffbf0e","last_reissued_at":"2026-08-04T00:30:17.559853Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T00:30:17.559853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Isaiah Siegl","submitted_at":"2025-09-02T21:20:53Z","abstract_excerpt":"Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are $e$-positive. Finding a combinatorial interpretation for these $e$-coefficients remains a major open problem. One approach is to look for combinatorial interpretations which are subsets of Gasharov's $P$-tableaux. Towards this goal, we introduce sets of strong and powerful $P$-tableaux, and use them to find combinatorial interpretations for various $e$-coefficients of the chromatic symmetric function $X_{inc(P)}(\\mathbf{x}, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.02841","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.02841/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2509.02841","created_at":"2026-08-04T00:30:17.560822+00:00"},{"alias_kind":"arxiv_version","alias_value":"2509.02841v3","created_at":"2026-08-04T00:30:17.560822+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.02841","created_at":"2026-08-04T00:30:17.560822+00:00"},{"alias_kind":"pith_short_12","alias_value":"XTUBC5D7TW3E","created_at":"2026-08-04T00:30:17.560822+00:00"},{"alias_kind":"pith_short_16","alias_value":"XTUBC5D7TW3EWVGO","created_at":"2026-08-04T00:30:17.560822+00:00"},{"alias_kind":"pith_short_8","alias_value":"XTUBC5D7","created_at":"2026-08-04T00:30:17.560822+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/XTUBC5D7TW3EWVGOQI3RZQFDCM","json":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM.json","graph_json":"https://pith.science/api/pith-number/XTUBC5D7TW3EWVGOQI3RZQFDCM/graph.json","events_json":"https://pith.science/api/pith-number/XTUBC5D7TW3EWVGOQI3RZQFDCM/events.json","paper":"https://pith.science/paper/XTUBC5D7"},"agent_actions":{"view_html":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM","download_json":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM.json","view_paper":"https://pith.science/paper/XTUBC5D7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2509.02841&json=true","fetch_graph":"https://pith.science/api/pith-number/XTUBC5D7TW3EWVGOQI3RZQFDCM/graph.json","fetch_events":"https://pith.science/api/pith-number/XTUBC5D7TW3EWVGOQI3RZQFDCM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM/action/storage_attestation","attest_author":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM/action/author_attestation","sign_citation":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM/action/citation_signature","submit_replication":"https://pith.science/pith/XTUBC5D7TW3EWVGOQI3RZQFDCM/action/replication_record"}},"created_at":"2026-08-04T00:30:17.560822+00:00","updated_at":"2026-08-04T00:30:17.560822+00:00"}