{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4EATX7PJJBAACGDQHZNOD6AAG3","short_pith_number":"pith:4EATX7PJ","canonical_record":{"source":{"id":"1510.05194","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-18T03:02:53Z","cross_cats_sorted":[],"title_canon_sha256":"c142f20fefe8d3d819b9ae249b01744eaf989027cfd0f33121c25a80b148b336","abstract_canon_sha256":"9b9b26208f6d8d7fc98a3d20791f52060e51d6781eb28234c0fce60f462ef190"},"schema_version":"1.0"},"canonical_sha256":"e1013bfde948400118703e5ae1f80036e3bca07c5838c78ac76db967a377c10e","source":{"kind":"arxiv","id":"1510.05194","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05194","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05194v3","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05194","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"pith_short_12","alias_value":"4EATX7PJJBAA","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4EATX7PJJBAACGDQ","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4EATX7PJ","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4EATX7PJJBAACGDQHZNOD6AAG3","target":"record","payload":{"canonical_record":{"source":{"id":"1510.05194","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-18T03:02:53Z","cross_cats_sorted":[],"title_canon_sha256":"c142f20fefe8d3d819b9ae249b01744eaf989027cfd0f33121c25a80b148b336","abstract_canon_sha256":"9b9b26208f6d8d7fc98a3d20791f52060e51d6781eb28234c0fce60f462ef190"},"schema_version":"1.0"},"canonical_sha256":"e1013bfde948400118703e5ae1f80036e3bca07c5838c78ac76db967a377c10e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:02.921463Z","signature_b64":"dgiNAo5khEPQd9ojxJj9KpD+UgNG1DnHoJcHVEA6xHGAYhIN71jfSNN/pKCN847Am3u3X6m6jMXA2o9MwnGKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1013bfde948400118703e5ae1f80036e3bca07c5838c78ac76db967a377c10e","last_reissued_at":"2026-05-18T00:28:02.920800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:02.920800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.05194","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":"2026-05-18T00:28:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3fpFveeu8SodXIn8FSum2zOaEmsH4Y/4jm3eZqRRsTAd1q2NM1hVrQ6dDk82gU7v0OLDXeHt6sbwyzfX4aL/Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:17:03.324280Z"},"content_sha256":"e76201adc4c2b5705600bde754b6575aac16df669f003a5dc042f668df80966e","schema_version":"1.0","event_id":"sha256:e76201adc4c2b5705600bde754b6575aac16df669f003a5dc042f668df80966e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4EATX7PJJBAACGDQHZNOD6AAG3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bijections for planar maps with boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"\\'Eric Fusy, Olivier Bernardi","submitted_at":"2015-10-18T03:02:53Z","abstract_excerpt":"We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05194","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":"2026-05-18T00:28:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n0G8UWpVV6353gQjcLyeAkHkjhZAVQd10WW/zH56PS4dta49yHL5vBqJ7sE3rPyHAZL4Rgn0aFHZyukRANd9DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:17:03.324885Z"},"content_sha256":"45a80db12b558a49d475d6a86346c7a19fe22cf89172b36c0cd356e76625d76a","schema_version":"1.0","event_id":"sha256:45a80db12b558a49d475d6a86346c7a19fe22cf89172b36c0cd356e76625d76a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4EATX7PJJBAACGDQHZNOD6AAG3/bundle.json","state_url":"https://pith.science/pith/4EATX7PJJBAACGDQHZNOD6AAG3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4EATX7PJJBAACGDQHZNOD6AAG3/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":"2026-06-06T21:17:03Z","links":{"resolver":"https://pith.science/pith/4EATX7PJJBAACGDQHZNOD6AAG3","bundle":"https://pith.science/pith/4EATX7PJJBAACGDQHZNOD6AAG3/bundle.json","state":"https://pith.science/pith/4EATX7PJJBAACGDQHZNOD6AAG3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4EATX7PJJBAACGDQHZNOD6AAG3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4EATX7PJJBAACGDQHZNOD6AAG3","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":"9b9b26208f6d8d7fc98a3d20791f52060e51d6781eb28234c0fce60f462ef190","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-18T03:02:53Z","title_canon_sha256":"c142f20fefe8d3d819b9ae249b01744eaf989027cfd0f33121c25a80b148b336"},"schema_version":"1.0","source":{"id":"1510.05194","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05194","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05194v3","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05194","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"pith_short_12","alias_value":"4EATX7PJJBAA","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4EATX7PJJBAACGDQ","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4EATX7PJ","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:45a80db12b558a49d475d6a86346c7a19fe22cf89172b36c0cd356e76625d76a","target":"graph","created_at":"2026-05-18T00:28:02Z","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 present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained","authors_text":"\\'Eric Fusy, Olivier Bernardi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-18T03:02:53Z","title":"Bijections for planar maps with boundaries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05194","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:e76201adc4c2b5705600bde754b6575aac16df669f003a5dc042f668df80966e","target":"record","created_at":"2026-05-18T00:28:02Z","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":"9b9b26208f6d8d7fc98a3d20791f52060e51d6781eb28234c0fce60f462ef190","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-18T03:02:53Z","title_canon_sha256":"c142f20fefe8d3d819b9ae249b01744eaf989027cfd0f33121c25a80b148b336"},"schema_version":"1.0","source":{"id":"1510.05194","kind":"arxiv","version":3}},"canonical_sha256":"e1013bfde948400118703e5ae1f80036e3bca07c5838c78ac76db967a377c10e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1013bfde948400118703e5ae1f80036e3bca07c5838c78ac76db967a377c10e","first_computed_at":"2026-05-18T00:28:02.920800Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:02.920800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dgiNAo5khEPQd9ojxJj9KpD+UgNG1DnHoJcHVEA6xHGAYhIN71jfSNN/pKCN847Am3u3X6m6jMXA2o9MwnGKBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:02.921463Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.05194","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e76201adc4c2b5705600bde754b6575aac16df669f003a5dc042f668df80966e","sha256:45a80db12b558a49d475d6a86346c7a19fe22cf89172b36c0cd356e76625d76a"],"state_sha256":"fd987824ac20d76106a272bcab06ec0dbe83ef691cc1cda11cc9255373ae2606"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CNS5WeGzav37KofQHMMA0deXYTACgGrnaztmhNQ1uJsQFxUgIN3Bi/eLVC4c9XTpG58IM5lTG1wWZfJDwMxGCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T21:17:03.328603Z","bundle_sha256":"ee46d8af4c2002a079b7a9a93f16c11dde14f816acd05cabe303198c7c2ac5d5"}}