{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QO2G4JYZE4P3WSYIQCGTJTZHER","short_pith_number":"pith:QO2G4JYZ","canonical_record":{"source":{"id":"1101.5608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-28T19:10:13Z","cross_cats_sorted":[],"title_canon_sha256":"3d8701cf36270b3c166668fb7214e094ef851418aa05f424caaece828d6b39e0","abstract_canon_sha256":"05b96d571790fad61dac17a68f3d23d26daa1c592a36f0ee03e2e3eac333e124"},"schema_version":"1.0"},"canonical_sha256":"83b46e2719271fbb4b08808d34cf2724619cd0cc5985a754befe0d2277afe8ba","source":{"kind":"arxiv","id":"1101.5608","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.5608","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"arxiv_version","alias_value":"1101.5608v1","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.5608","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"pith_short_12","alias_value":"QO2G4JYZE4P3","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QO2G4JYZE4P3WSYI","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QO2G4JYZ","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QO2G4JYZE4P3WSYIQCGTJTZHER","target":"record","payload":{"canonical_record":{"source":{"id":"1101.5608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-28T19:10:13Z","cross_cats_sorted":[],"title_canon_sha256":"3d8701cf36270b3c166668fb7214e094ef851418aa05f424caaece828d6b39e0","abstract_canon_sha256":"05b96d571790fad61dac17a68f3d23d26daa1c592a36f0ee03e2e3eac333e124"},"schema_version":"1.0"},"canonical_sha256":"83b46e2719271fbb4b08808d34cf2724619cd0cc5985a754befe0d2277afe8ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:36.450924Z","signature_b64":"r3bAV7KIhjyBfD/s+MfKujjkDrXm0W1epEjUcKPnjnjFMqFdAYOLhY2oxNuTdoYe9plo2t5yi35ZVNxXnj0rCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83b46e2719271fbb4b08808d34cf2724619cd0cc5985a754befe0d2277afe8ba","last_reissued_at":"2026-05-18T00:35:36.450311Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:36.450311Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.5608","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:35:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LFrZ0ssWWud+h2R5O1EVIuCUl06huraFEFDwr4j0coO4M3i1KezG+Ehi45mMVc7me6joQGc/bxXzPbl2kGIkDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:10:00.524504Z"},"content_sha256":"c59a2981f30e6c158341a2dded4863fc24f88cfab3965593bd403e58b471346b","schema_version":"1.0","event_id":"sha256:c59a2981f30e6c158341a2dded4863fc24f88cfab3965593bd403e58b471346b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QO2G4JYZE4P3WSYIQCGTJTZHER","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jang Soo Kim, Matthieu Josuat-Verg\\`es","submitted_at":"2011-01-28T19:10:13Z","abstract_excerpt":"Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard-Riordan--like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5608","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":""},"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":"2026-05-18T00:35:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qQor2nJbccXBcuq2Cn0YWKKWY46Dz9AqMaBMmcdjXdVhRNMntDcc4JaFthoz0wafvBLaC3BdSGpasyQkjddIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:10:00.525444Z"},"content_sha256":"41d9fa87951515fb968bd871dd7881df81676797bdb7618e6f691a77a7e22678","schema_version":"1.0","event_id":"sha256:41d9fa87951515fb968bd871dd7881df81676797bdb7618e6f691a77a7e22678"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/bundle.json","state_url":"https://pith.science/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/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":"2026-05-25T17:10:00Z","links":{"resolver":"https://pith.science/pith/QO2G4JYZE4P3WSYIQCGTJTZHER","bundle":"https://pith.science/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/bundle.json","state":"https://pith.science/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QO2G4JYZE4P3WSYIQCGTJTZHER/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QO2G4JYZE4P3WSYIQCGTJTZHER","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":"05b96d571790fad61dac17a68f3d23d26daa1c592a36f0ee03e2e3eac333e124","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-28T19:10:13Z","title_canon_sha256":"3d8701cf36270b3c166668fb7214e094ef851418aa05f424caaece828d6b39e0"},"schema_version":"1.0","source":{"id":"1101.5608","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.5608","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"arxiv_version","alias_value":"1101.5608v1","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.5608","created_at":"2026-05-18T00:35:36Z"},{"alias_kind":"pith_short_12","alias_value":"QO2G4JYZE4P3","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QO2G4JYZE4P3WSYI","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QO2G4JYZ","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:41d9fa87951515fb968bd871dd7881df81676797bdb7618e6f691a77a7e22678","target":"graph","created_at":"2026-05-18T00:35:36Z","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":"Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard-Riordan--like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between","authors_text":"Jang Soo Kim, Matthieu Josuat-Verg\\`es","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-28T19:10:13Z","title":"Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5608","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:c59a2981f30e6c158341a2dded4863fc24f88cfab3965593bd403e58b471346b","target":"record","created_at":"2026-05-18T00:35:36Z","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":"05b96d571790fad61dac17a68f3d23d26daa1c592a36f0ee03e2e3eac333e124","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-28T19:10:13Z","title_canon_sha256":"3d8701cf36270b3c166668fb7214e094ef851418aa05f424caaece828d6b39e0"},"schema_version":"1.0","source":{"id":"1101.5608","kind":"arxiv","version":1}},"canonical_sha256":"83b46e2719271fbb4b08808d34cf2724619cd0cc5985a754befe0d2277afe8ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"83b46e2719271fbb4b08808d34cf2724619cd0cc5985a754befe0d2277afe8ba","first_computed_at":"2026-05-18T00:35:36.450311Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:36.450311Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r3bAV7KIhjyBfD/s+MfKujjkDrXm0W1epEjUcKPnjnjFMqFdAYOLhY2oxNuTdoYe9plo2t5yi35ZVNxXnj0rCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:36.450924Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.5608","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c59a2981f30e6c158341a2dded4863fc24f88cfab3965593bd403e58b471346b","sha256:41d9fa87951515fb968bd871dd7881df81676797bdb7618e6f691a77a7e22678"],"state_sha256":"2566a9d13052d98e55e25816f9986129b9226855a6a464555ff695b9126fe47d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kov101CaTYeBeZGKgq3sJ5lqX4ssD/ktVeOXL6a0/iKtq+cf/3ra6kIzPU63/ynNOryK5hMAnmsAxhIkE3dCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T17:10:00.529123Z","bundle_sha256":"b0779e9dd8fd95b13a11a620b42c0e4ec634da96d67e67744950b41c427e6bd3"}}