{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DO4BEL3HFOOZYN6ZHWYB3ZNS2A","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":"f5f38fe753a359e7ca601f03d70c1dd4b04e6dc23658f0440c56141787ae5b5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-18T09:06:52Z","title_canon_sha256":"b8f6ec48118041848098bad6ec4173ccbba20f6f2fc79f90f49cb76408401e04"},"schema_version":"1.0","source":{"id":"1810.07956","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.07956","created_at":"2026-05-18T00:02:53Z"},{"alias_kind":"arxiv_version","alias_value":"1810.07956v1","created_at":"2026-05-18T00:02:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07956","created_at":"2026-05-18T00:02:53Z"},{"alias_kind":"pith_short_12","alias_value":"DO4BEL3HFOOZ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DO4BEL3HFOOZYN6Z","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DO4BEL3H","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:d17c76ad2c1d6bfb5a8a1b4119b5b985749195f40462a84d442e8948cdb6b4c7","target":"graph","created_at":"2026-05-18T00:02:53Z","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":"In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely, \\begin{eqnarray*} A_n(p,q)=\\sum_{\\pi\\in \\mathfrak{S}_n}p^{{\\rm odes}(\\pi)}q^{{\\rm edes}(\\pi)} \\end{eqnarray*} for $n\\geq 1$, where ${odes}(\\pi)$ and ${edes}(\\pi)$ enumerate the number of descents of permutation $\\pi$ in odd and even positions, respectively. In this paper, we build an exponential generating function for $A_{n}(p,q)$ and establish an explicit formula for $A_{n}(p,q)$ in terms of Eulerian polynomials $A_{n}(q)$ and $C(q)$, the generating function for Catalan numbers. In certain special case, we set up a","authors_text":"Liting Zhai, Yidong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-18T09:06:52Z","title":"Some properties of a class of refined Eulerian polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07956","kind":"arxiv","version":1},"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":"sha256:14bc78d49c3a313e3fdd1267d0e8d4b538f07b7f18684d15d71344c4107c8097","target":"record","created_at":"2026-05-18T00:02:53Z","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":"f5f38fe753a359e7ca601f03d70c1dd4b04e6dc23658f0440c56141787ae5b5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-18T09:06:52Z","title_canon_sha256":"b8f6ec48118041848098bad6ec4173ccbba20f6f2fc79f90f49cb76408401e04"},"schema_version":"1.0","source":{"id":"1810.07956","kind":"arxiv","version":1}},"canonical_sha256":"1bb8122f672b9d9c37d93db01de5b2d00b047f58ff82054a51601a24e607fb7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bb8122f672b9d9c37d93db01de5b2d00b047f58ff82054a51601a24e607fb7d","first_computed_at":"2026-05-18T00:02:53.647555Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:53.647555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LqOuHOKvX05sbB8yoXLMrWdvnYV06PSn/fjvFMI3aMINta4SH8vYnGcqiYmQCW/dPlqg+l1OijDMQZ0NrzSHDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:53.648201Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.07956","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14bc78d49c3a313e3fdd1267d0e8d4b538f07b7f18684d15d71344c4107c8097","sha256:d17c76ad2c1d6bfb5a8a1b4119b5b985749195f40462a84d442e8948cdb6b4c7"],"state_sha256":"e783d2868a33ea18e23830f165f73019fdd63afc3cbd5a4b7169494267370a65"}