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Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$ \\sum_{k=0}^{n-1}I_{n,k}t^k=\\sum_{k=0}^{\\lfloor (n-1)/2\\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}. $$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right."},"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":"math/0504195","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2005-04-10T16:43:36Z","cross_cats_sorted":[],"title_canon_sha256":"5e6777577a99aa67b21055f0095e3c394507acaf51267f1c817d3d5fec9e104c","abstract_canon_sha256":"222b1dab68c5ddae33a4b1a2b6ab6c5d3bf85eaade27026a4015ed0b43e688e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:00.768280Z","signature_b64":"4XPn7GQemhU3kliQbEQ0gaBNkU0jLCyvt32NcEc03m+HLLCG+RHyEqW84VMyJjK72Bex/jIE7UXpiv2PqUncAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b86b1e30b72c8c039ee58505aca92f0e207d4da3c8998004614283f74142b3de","last_reissued_at":"2026-05-18T04:26:00.767913Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:00.767913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Eulerian Distribution on Involutions is Indeed Unimodal","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiang Zeng, Victor J. 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Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$ \\sum_{k=0}^{n-1}I_{n,k}t^k=\\sum_{k=0}^{\\lfloor (n-1)/2\\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}. $$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0504195","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":""},"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":"math/0504195","created_at":"2026-05-18T04:26:00.767968+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0504195v3","created_at":"2026-05-18T04:26:00.767968+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0504195","created_at":"2026-05-18T04:26:00.767968+00:00"},{"alias_kind":"pith_short_12","alias_value":"XBVR4MFXFSGA","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"XBVR4MFXFSGAHHXF","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"XBVR4MFX","created_at":"2026-05-18T12:25:53.939244+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/XBVR4MFXFSGAHHXFQUC2ZKJPBY","json":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY.json","graph_json":"https://pith.science/api/pith-number/XBVR4MFXFSGAHHXFQUC2ZKJPBY/graph.json","events_json":"https://pith.science/api/pith-number/XBVR4MFXFSGAHHXFQUC2ZKJPBY/events.json","paper":"https://pith.science/paper/XBVR4MFX"},"agent_actions":{"view_html":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY","download_json":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY.json","view_paper":"https://pith.science/paper/XBVR4MFX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0504195&json=true","fetch_graph":"https://pith.science/api/pith-number/XBVR4MFXFSGAHHXFQUC2ZKJPBY/graph.json","fetch_events":"https://pith.science/api/pith-number/XBVR4MFXFSGAHHXFQUC2ZKJPBY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY/action/storage_attestation","attest_author":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY/action/author_attestation","sign_citation":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY/action/citation_signature","submit_replication":"https://pith.science/pith/XBVR4MFXFSGAHHXFQUC2ZKJPBY/action/replication_record"}},"created_at":"2026-05-18T04:26:00.767968+00:00","updated_at":"2026-05-18T04:26:00.767968+00:00"}