{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:UGDE445KRFQSZ3M3U6RCEOQTUX","short_pith_number":"pith:UGDE445K","schema_version":"1.0","canonical_sha256":"a1864e73aa89612ced9ba7a2223a13a5c6390a0d3b4dcaab8328c9d868f5d50e","source":{"kind":"arxiv","id":"1211.6359","version":2},"attestation_state":"computed","paper":{"title":"On some generalized $q$-Eulerian polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhicong Lin","submitted_at":"2012-11-27T16:55:49Z","abstract_excerpt":"The $(q,r)$-Eulerian polynomials are the $(\\maj-$$\\exc,\\fix,\\exc)$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic $(\\inv-$$\\lec,\\pix,\\lec)$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted descent polynomials. In particular, we obtain a generalized symmetrical id"},"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":"1211.6359","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-27T16:55:49Z","cross_cats_sorted":[],"title_canon_sha256":"a73ee6fd3a11e75d49e6160e26bb8dfee32914a6b9cb23deed9b43df5af0ede0","abstract_canon_sha256":"8b044087412891351f258cb1ed8831e481caa60756cad23fd96573beb5478809"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:29.205312Z","signature_b64":"wI/6mQHAVyRTMcjhMRZR6NuJdQO6hxGVyJJCKwqsRLUMJ4UWnpfeBVL/0FFhh0CVb9/aWxIWQNjrcjp2uCCzBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1864e73aa89612ced9ba7a2223a13a5c6390a0d3b4dcaab8328c9d868f5d50e","last_reissued_at":"2026-05-18T03:31:29.204818Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:29.204818Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some generalized $q$-Eulerian polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhicong Lin","submitted_at":"2012-11-27T16:55:49Z","abstract_excerpt":"The $(q,r)$-Eulerian polynomials are the $(\\maj-$$\\exc,\\fix,\\exc)$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic $(\\inv-$$\\lec,\\pix,\\lec)$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted descent polynomials. In particular, we obtain a generalized symmetrical id"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6359","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":"1211.6359","created_at":"2026-05-18T03:31:29.204896+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.6359v2","created_at":"2026-05-18T03:31:29.204896+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6359","created_at":"2026-05-18T03:31:29.204896+00:00"},{"alias_kind":"pith_short_12","alias_value":"UGDE445KRFQS","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"UGDE445KRFQSZ3M3","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"UGDE445K","created_at":"2026-05-18T12:27:23.164592+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/UGDE445KRFQSZ3M3U6RCEOQTUX","json":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX.json","graph_json":"https://pith.science/api/pith-number/UGDE445KRFQSZ3M3U6RCEOQTUX/graph.json","events_json":"https://pith.science/api/pith-number/UGDE445KRFQSZ3M3U6RCEOQTUX/events.json","paper":"https://pith.science/paper/UGDE445K"},"agent_actions":{"view_html":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX","download_json":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX.json","view_paper":"https://pith.science/paper/UGDE445K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.6359&json=true","fetch_graph":"https://pith.science/api/pith-number/UGDE445KRFQSZ3M3U6RCEOQTUX/graph.json","fetch_events":"https://pith.science/api/pith-number/UGDE445KRFQSZ3M3U6RCEOQTUX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX/action/storage_attestation","attest_author":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX/action/author_attestation","sign_citation":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX/action/citation_signature","submit_replication":"https://pith.science/pith/UGDE445KRFQSZ3M3U6RCEOQTUX/action/replication_record"}},"created_at":"2026-05-18T03:31:29.204896+00:00","updated_at":"2026-05-18T03:31:29.204896+00:00"}