{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:64YSQCG4KKF2D4LMF7B4XFRZOD","short_pith_number":"pith:64YSQCG4","schema_version":"1.0","canonical_sha256":"f7312808dc528ba1f16c2fc3cb963970ca16946f500bc8dfb04c97eccdde5777","source":{"kind":"arxiv","id":"1908.03914","version":1},"attestation_state":"computed","paper":{"title":"Arithmetic of weighted Catalan numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Andrew Gu, Yibo Gao","submitted_at":"2019-08-11T14:30:48Z","abstract_excerpt":"In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to $2$-adic integers. These methods are also extended to $q$-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to t"},"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":"1908.03914","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-11T14:30:48Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"54fd22d1a0ef716e8f10d9a0993b1b5d3e29eeeded060cdad12e9cab1825bde8","abstract_canon_sha256":"6e814ff3bea00c4cfd73e8874e73d20c7f6ea44c821deddcc61d280b9f45667e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:53:19.344767Z","signature_b64":"mF2xGqKtKnSEOG09dCUlvXAtm4jNgyJGBPb7qWbT2Kc01byqvGgWeWACguOhtg7ria+VnVnipiIaWOONcT/9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7312808dc528ba1f16c2fc3cb963970ca16946f500bc8dfb04c97eccdde5777","last_reissued_at":"2026-07-04T23:53:19.344407Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:53:19.344407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic of weighted Catalan numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Andrew Gu, Yibo Gao","submitted_at":"2019-08-11T14:30:48Z","abstract_excerpt":"In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to $2$-adic integers. These methods are also extended to $q$-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03914","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03914/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.03914","created_at":"2026-07-04T23:53:19.344463+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03914v1","created_at":"2026-07-04T23:53:19.344463+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03914","created_at":"2026-07-04T23:53:19.344463+00:00"},{"alias_kind":"pith_short_12","alias_value":"64YSQCG4KKF2","created_at":"2026-07-04T23:53:19.344463+00:00"},{"alias_kind":"pith_short_16","alias_value":"64YSQCG4KKF2D4LM","created_at":"2026-07-04T23:53:19.344463+00:00"},{"alias_kind":"pith_short_8","alias_value":"64YSQCG4","created_at":"2026-07-04T23:53:19.344463+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/64YSQCG4KKF2D4LMF7B4XFRZOD","json":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD.json","graph_json":"https://pith.science/api/pith-number/64YSQCG4KKF2D4LMF7B4XFRZOD/graph.json","events_json":"https://pith.science/api/pith-number/64YSQCG4KKF2D4LMF7B4XFRZOD/events.json","paper":"https://pith.science/paper/64YSQCG4"},"agent_actions":{"view_html":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD","download_json":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD.json","view_paper":"https://pith.science/paper/64YSQCG4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03914&json=true","fetch_graph":"https://pith.science/api/pith-number/64YSQCG4KKF2D4LMF7B4XFRZOD/graph.json","fetch_events":"https://pith.science/api/pith-number/64YSQCG4KKF2D4LMF7B4XFRZOD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD/action/storage_attestation","attest_author":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD/action/author_attestation","sign_citation":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD/action/citation_signature","submit_replication":"https://pith.science/pith/64YSQCG4KKF2D4LMF7B4XFRZOD/action/replication_record"}},"created_at":"2026-07-04T23:53:19.344463+00:00","updated_at":"2026-07-04T23:53:19.344463+00:00"}