{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:R4RWZZHE5WZKY6TKYYNIZD752E","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":"bd7a254f01278194a5a79b4c8ce3e9e237b26cd63dcf2cb361c70b0d1b3e9fcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-25T23:14:27Z","title_canon_sha256":"eb112d055825109cb6064a866d54da395d9bd82cf98ca74fbb7e2c4ce6fe736d"},"schema_version":"1.0","source":{"id":"1808.08481","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.08481","created_at":"2026-05-17T23:53:50Z"},{"alias_kind":"arxiv_version","alias_value":"1808.08481v2","created_at":"2026-05-17T23:53:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.08481","created_at":"2026-05-17T23:53:50Z"},{"alias_kind":"pith_short_12","alias_value":"R4RWZZHE5WZK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"R4RWZZHE5WZKY6TK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"R4RWZZHE","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:920bc35c033456a87576ec79c4b1c50be4ada144fbbb3dc2077765d4d53b0a83","target":"graph","created_at":"2026-05-17T23:53:50Z","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":"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 ","authors_text":"Danielle Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-25T23:14:27Z","title":"The Eulerian distribution on involutions is indeed $\\gamma$-positive"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08481","kind":"arxiv","version":2},"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:cf83daf6850f180884c6f75b7380b080df2660ecd061925b5ff395a7e275cdf9","target":"record","created_at":"2026-05-17T23:53:50Z","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":"bd7a254f01278194a5a79b4c8ce3e9e237b26cd63dcf2cb361c70b0d1b3e9fcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-25T23:14:27Z","title_canon_sha256":"eb112d055825109cb6064a866d54da395d9bd82cf98ca74fbb7e2c4ce6fe736d"},"schema_version":"1.0","source":{"id":"1808.08481","kind":"arxiv","version":2}},"canonical_sha256":"8f236ce4e4edb2ac7a6ac61a8c8ffdd121c2cb26114d5dcad76e2d3389d8d748","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f236ce4e4edb2ac7a6ac61a8c8ffdd121c2cb26114d5dcad76e2d3389d8d748","first_computed_at":"2026-05-17T23:53:50.923915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:50.923915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c5QKMwEfGqUL1zgUp6oZdC2t39HSPm9ZZUHG1kyKyLOTScVZ4uHD+cwbot5cLUsexBXhXxFON0rnnJ0VUiYIDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:50.924554Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.08481","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf83daf6850f180884c6f75b7380b080df2660ecd061925b5ff395a7e275cdf9","sha256:920bc35c033456a87576ec79c4b1c50be4ada144fbbb3dc2077765d4d53b0a83"],"state_sha256":"0c64a4b20a2e550252d288b0e32e70b6dd0af85d4469387cbcda88d5bd42b5a3"}