{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:Y5OEHNOGVOKRIO6AEVYAOBQTBX","short_pith_number":"pith:Y5OEHNOG","canonical_record":{"source":{"id":"1104.4584","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-23T18:51:51Z","cross_cats_sorted":[],"title_canon_sha256":"03b0e0a11b3dd4da1002ae42cacea1a4366c81ca0339f94dddd15041afdf9494","abstract_canon_sha256":"4a35aae0dd1a8a8b905646859fd5d704a473f06f34f5b74ac9b414b16680fb8c"},"schema_version":"1.0"},"canonical_sha256":"c75c43b5c6ab95143bc025700706130dd228e4bf34a624faeae35e62905be14f","source":{"kind":"arxiv","id":"1104.4584","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4584","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4584v3","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4584","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"Y5OEHNOGVOKR","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Y5OEHNOGVOKRIO6A","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Y5OEHNOG","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:Y5OEHNOGVOKRIO6AEVYAOBQTBX","target":"record","payload":{"canonical_record":{"source":{"id":"1104.4584","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-23T18:51:51Z","cross_cats_sorted":[],"title_canon_sha256":"03b0e0a11b3dd4da1002ae42cacea1a4366c81ca0339f94dddd15041afdf9494","abstract_canon_sha256":"4a35aae0dd1a8a8b905646859fd5d704a473f06f34f5b74ac9b414b16680fb8c"},"schema_version":"1.0"},"canonical_sha256":"c75c43b5c6ab95143bc025700706130dd228e4bf34a624faeae35e62905be14f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:46.296811Z","signature_b64":"hG+XHCiBq4h2etbf8OJ5tiCShNCJTCQSOHcLJYxZbO4mrzFp/bVPb1KsYzCka0H9ZuhUlPy9dmKy4VmA+luXDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c75c43b5c6ab95143bc025700706130dd228e4bf34a624faeae35e62905be14f","last_reissued_at":"2026-05-18T03:42:46.296230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:46.296230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.4584","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fm/x0yl5/XwYDGx/kAFuKzA63Ec3/5HXXCZFPFKLizti5LdeKYo648c753m1Z9gdp9cCBxwQGBNGRpDh+jdbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:41:34.915513Z"},"content_sha256":"6992f74553b21551ef484ed8c5f8f8255dea1d51397fce3b2125176bbc79532a","schema_version":"1.0","event_id":"sha256:6992f74553b21551ef484ed8c5f8f8255dea1d51397fce3b2125176bbc79532a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:Y5OEHNOGVOKRIO6AEVYAOBQTBX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Enumeration formulas for generalized q-Euler numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jang Soo Kim","submitted_at":"2011-04-23T18:51:51Z","abstract_excerpt":"We find an enumeration formula for a $(t,q)$-Euler number which is a generalization of the $q$-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the $(t,q)$-Euler number and find another formula when $t=\\pm q^r$ for any integer $r$. Special cases of our latter formula include the formula of the $q$-Euler number recently found by Josuat-Verg\\`es and Touchard-Riordan's formula."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4584","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pztNOQF8zl8+7CaSoVLmbmeESaa/aX2pby2PybxQwS7MOygVHSHu/+Iw5xeX5nA5L988cgBowd00UFEioJNTDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:41:34.915879Z"},"content_sha256":"c90e12f5c7a87eb65d024b2d0c17cccba136d344d47a1c94ae8e3e6afff67293","schema_version":"1.0","event_id":"sha256:c90e12f5c7a87eb65d024b2d0c17cccba136d344d47a1c94ae8e3e6afff67293"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/bundle.json","state_url":"https://pith.science/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T23:41:34Z","links":{"resolver":"https://pith.science/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX","bundle":"https://pith.science/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/bundle.json","state":"https://pith.science/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y5OEHNOGVOKRIO6AEVYAOBQTBX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:Y5OEHNOGVOKRIO6AEVYAOBQTBX","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":"4a35aae0dd1a8a8b905646859fd5d704a473f06f34f5b74ac9b414b16680fb8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-23T18:51:51Z","title_canon_sha256":"03b0e0a11b3dd4da1002ae42cacea1a4366c81ca0339f94dddd15041afdf9494"},"schema_version":"1.0","source":{"id":"1104.4584","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4584","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4584v3","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4584","created_at":"2026-05-18T03:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"Y5OEHNOGVOKR","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Y5OEHNOGVOKRIO6A","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Y5OEHNOG","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:c90e12f5c7a87eb65d024b2d0c17cccba136d344d47a1c94ae8e3e6afff67293","target":"graph","created_at":"2026-05-18T03:42:46Z","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":"We find an enumeration formula for a $(t,q)$-Euler number which is a generalization of the $q$-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the $(t,q)$-Euler number and find another formula when $t=\\pm q^r$ for any integer $r$. Special cases of our latter formula include the formula of the $q$-Euler number recently found by Josuat-Verg\\`es and Touchard-Riordan's formula.","authors_text":"Jang Soo Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-23T18:51:51Z","title":"Enumeration formulas for generalized q-Euler numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4584","kind":"arxiv","version":3},"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":"sha256:6992f74553b21551ef484ed8c5f8f8255dea1d51397fce3b2125176bbc79532a","target":"record","created_at":"2026-05-18T03:42:46Z","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":"4a35aae0dd1a8a8b905646859fd5d704a473f06f34f5b74ac9b414b16680fb8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-23T18:51:51Z","title_canon_sha256":"03b0e0a11b3dd4da1002ae42cacea1a4366c81ca0339f94dddd15041afdf9494"},"schema_version":"1.0","source":{"id":"1104.4584","kind":"arxiv","version":3}},"canonical_sha256":"c75c43b5c6ab95143bc025700706130dd228e4bf34a624faeae35e62905be14f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c75c43b5c6ab95143bc025700706130dd228e4bf34a624faeae35e62905be14f","first_computed_at":"2026-05-18T03:42:46.296230Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:46.296230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hG+XHCiBq4h2etbf8OJ5tiCShNCJTCQSOHcLJYxZbO4mrzFp/bVPb1KsYzCka0H9ZuhUlPy9dmKy4VmA+luXDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:46.296811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.4584","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6992f74553b21551ef484ed8c5f8f8255dea1d51397fce3b2125176bbc79532a","sha256:c90e12f5c7a87eb65d024b2d0c17cccba136d344d47a1c94ae8e3e6afff67293"],"state_sha256":"f33b4494229e034c61b112fec3b72613a8bc0a6b4b73834bf5de753f2119663e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5MMVUhiY3FRhBrF5T5yoYD7i8E7L4g2FH2BFSLN0WSLqgvRv+K8fCG0kflJb8wm8Zj0TuJ7IOefWT5TzGaDHAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T23:41:34.918397Z","bundle_sha256":"0e2bd880b7749a62100451ddb20424c71cef5d3efc8831c9d272be747b814819"}}