{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:R4RWZZHE5WZKY6TKYYNIZD752E","short_pith_number":"pith:R4RWZZHE","schema_version":"1.0","canonical_sha256":"8f236ce4e4edb2ac7a6ac61a8c8ffdd121c2cb26114d5dcad76e2d3389d8d748","source":{"kind":"arxiv","id":"1808.08481","version":2},"attestation_state":"computed","paper":{"title":"The Eulerian distribution on involutions is indeed $\\gamma$-positive","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danielle Wang","submitted_at":"2018-08-25T23:14:27Z","abstract_excerpt":"Let $\\mathcal I_n$ and $\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\{1, \\dots, n\\}$, respectively, and let $\\text{des}(\\pi)$ denote the number of descents of the permutation $\\pi$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\sum_{\\pi \\in \\mathcal I_n} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 1$ and $J_{2n}(t) := \\sum_{\\pi \\in \\mathcal J_{2n}} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to "},"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":"1808.08481","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-25T23:14:27Z","cross_cats_sorted":[],"title_canon_sha256":"eb112d055825109cb6064a866d54da395d9bd82cf98ca74fbb7e2c4ce6fe736d","abstract_canon_sha256":"bd7a254f01278194a5a79b4c8ce3e9e237b26cd63dcf2cb361c70b0d1b3e9fcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:50.924554Z","signature_b64":"c5QKMwEfGqUL1zgUp6oZdC2t39HSPm9ZZUHG1kyKyLOTScVZ4uHD+cwbot5cLUsexBXhXxFON0rnnJ0VUiYIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f236ce4e4edb2ac7a6ac61a8c8ffdd121c2cb26114d5dcad76e2d3389d8d748","last_reissued_at":"2026-05-17T23:53:50.923915Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:50.923915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Eulerian distribution on involutions is indeed $\\gamma$-positive","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danielle Wang","submitted_at":"2018-08-25T23:14:27Z","abstract_excerpt":"Let $\\mathcal I_n$ and $\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\{1, \\dots, n\\}$, respectively, and let $\\text{des}(\\pi)$ denote the number of descents of the permutation $\\pi$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\sum_{\\pi \\in \\mathcal I_n} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 1$ and $J_{2n}(t) := \\sum_{\\pi \\in \\mathcal J_{2n}} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08481","kind":"arxiv","version":2},"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":"1808.08481","created_at":"2026-05-17T23:53:50.924016+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.08481v2","created_at":"2026-05-17T23:53:50.924016+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.08481","created_at":"2026-05-17T23:53:50.924016+00:00"},{"alias_kind":"pith_short_12","alias_value":"R4RWZZHE5WZK","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"R4RWZZHE5WZKY6TK","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"R4RWZZHE","created_at":"2026-05-18T12:32:50.500415+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/R4RWZZHE5WZKY6TKYYNIZD752E","json":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E.json","graph_json":"https://pith.science/api/pith-number/R4RWZZHE5WZKY6TKYYNIZD752E/graph.json","events_json":"https://pith.science/api/pith-number/R4RWZZHE5WZKY6TKYYNIZD752E/events.json","paper":"https://pith.science/paper/R4RWZZHE"},"agent_actions":{"view_html":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E","download_json":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E.json","view_paper":"https://pith.science/paper/R4RWZZHE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.08481&json=true","fetch_graph":"https://pith.science/api/pith-number/R4RWZZHE5WZKY6TKYYNIZD752E/graph.json","fetch_events":"https://pith.science/api/pith-number/R4RWZZHE5WZKY6TKYYNIZD752E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E/action/storage_attestation","attest_author":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E/action/author_attestation","sign_citation":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E/action/citation_signature","submit_replication":"https://pith.science/pith/R4RWZZHE5WZKY6TKYYNIZD752E/action/replication_record"}},"created_at":"2026-05-17T23:53:50.924016+00:00","updated_at":"2026-05-17T23:53:50.924016+00:00"}