{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:IOCQNNVFV3FAR22H3LK34GUE5L","short_pith_number":"pith:IOCQNNVF","canonical_record":{"source":{"id":"1609.07593","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-09-24T09:50:25Z","cross_cats_sorted":["cs.DM","math.CO","math.LO"],"title_canon_sha256":"39854c80cf2227f403ce8eaeb43af558427e539a8f1ac7a9431bbc8062aa5356","abstract_canon_sha256":"bbf8c976f9c464bb80941af748fb955fa57a4a7e0d676fe44e6e444cf8c12114"},"schema_version":"1.0"},"canonical_sha256":"438506b6a5aeca08eb47dad5be1a84eae8be91a9b67a3442cf577a1bb1203843","source":{"kind":"arxiv","id":"1609.07593","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07593","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07593v2","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07593","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"pith_short_12","alias_value":"IOCQNNVFV3FA","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IOCQNNVFV3FAR22H","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IOCQNNVF","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:IOCQNNVFV3FAR22H3LK34GUE5L","target":"record","payload":{"canonical_record":{"source":{"id":"1609.07593","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-09-24T09:50:25Z","cross_cats_sorted":["cs.DM","math.CO","math.LO"],"title_canon_sha256":"39854c80cf2227f403ce8eaeb43af558427e539a8f1ac7a9431bbc8062aa5356","abstract_canon_sha256":"bbf8c976f9c464bb80941af748fb955fa57a4a7e0d676fe44e6e444cf8c12114"},"schema_version":"1.0"},"canonical_sha256":"438506b6a5aeca08eb47dad5be1a84eae8be91a9b67a3442cf577a1bb1203843","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:20.434230Z","signature_b64":"JbBC5vxNdI8Z8YJ4HFOQE7nJfOPS9GsSSk6hNiRAVkbVawSQ67V/V+Mk++WNtAnTCz58MOwjfkbz9A8B/najCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"438506b6a5aeca08eb47dad5be1a84eae8be91a9b67a3442cf577a1bb1203843","last_reissued_at":"2026-05-18T01:02:20.433569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:20.433569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.07593","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:02:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j2luKoKv7GhdiVwqjcyqz5ExdnaJ/VoeclaGVVcE8HVMmsD8XfnyaVEHr4uJzfJgo7tP5qvjudGi+W+MIYm5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T08:29:05.723160Z"},"content_sha256":"bf729f616d7ed11fec24f32cea96b206349cd497fc2eae3c473d983bd222ee62","schema_version":"1.0","event_id":"sha256:bf729f616d7ed11fec24f32cea96b206349cd497fc2eae3c473d983bd222ee62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:IOCQNNVFV3FAR22H3LK34GUE5L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorics of $\\lambda$-terms: a natural approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.LO"],"primary_cat":"cs.LO","authors_text":"Katarzyna Grygiel, Maciej Bendkowski, Marek Zaionc, Pierre Lescanne","submitted_at":"2016-09-24T09:50:25Z","abstract_excerpt":"We consider combinatorial aspects of $\\lambda$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $\\lambda$-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral $\\lambda$-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07593","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:02:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7tT48w+s/1VkgIHSNxdI/UWmw5F19yR0GnsADk532IE+h/oSTvS4fbDrYpsgE0gVgyanURHNhpVwjnyAA8qFBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T08:29:05.723523Z"},"content_sha256":"f2f4a2222f9dd4d6e5b3108c1521797e35c9145ace68b39b964eaadefea0eb8a","schema_version":"1.0","event_id":"sha256:f2f4a2222f9dd4d6e5b3108c1521797e35c9145ace68b39b964eaadefea0eb8a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IOCQNNVFV3FAR22H3LK34GUE5L/bundle.json","state_url":"https://pith.science/pith/IOCQNNVFV3FAR22H3LK34GUE5L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IOCQNNVFV3FAR22H3LK34GUE5L/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":"2026-06-12T08:29:05Z","links":{"resolver":"https://pith.science/pith/IOCQNNVFV3FAR22H3LK34GUE5L","bundle":"https://pith.science/pith/IOCQNNVFV3FAR22H3LK34GUE5L/bundle.json","state":"https://pith.science/pith/IOCQNNVFV3FAR22H3LK34GUE5L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IOCQNNVFV3FAR22H3LK34GUE5L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IOCQNNVFV3FAR22H3LK34GUE5L","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":"bbf8c976f9c464bb80941af748fb955fa57a4a7e0d676fe44e6e444cf8c12114","cross_cats_sorted":["cs.DM","math.CO","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-09-24T09:50:25Z","title_canon_sha256":"39854c80cf2227f403ce8eaeb43af558427e539a8f1ac7a9431bbc8062aa5356"},"schema_version":"1.0","source":{"id":"1609.07593","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07593","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07593v2","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07593","created_at":"2026-05-18T01:02:20Z"},{"alias_kind":"pith_short_12","alias_value":"IOCQNNVFV3FA","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IOCQNNVFV3FAR22H","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IOCQNNVF","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:f2f4a2222f9dd4d6e5b3108c1521797e35c9145ace68b39b964eaadefea0eb8a","target":"graph","created_at":"2026-05-18T01:02:20Z","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 consider combinatorial aspects of $\\lambda$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $\\lambda$-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral $\\lambda$-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size","authors_text":"Katarzyna Grygiel, Maciej Bendkowski, Marek Zaionc, Pierre Lescanne","cross_cats":["cs.DM","math.CO","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-09-24T09:50:25Z","title":"Combinatorics of $\\lambda$-terms: a natural approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07593","kind":"arxiv","version":2},"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:bf729f616d7ed11fec24f32cea96b206349cd497fc2eae3c473d983bd222ee62","target":"record","created_at":"2026-05-18T01:02:20Z","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":"bbf8c976f9c464bb80941af748fb955fa57a4a7e0d676fe44e6e444cf8c12114","cross_cats_sorted":["cs.DM","math.CO","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-09-24T09:50:25Z","title_canon_sha256":"39854c80cf2227f403ce8eaeb43af558427e539a8f1ac7a9431bbc8062aa5356"},"schema_version":"1.0","source":{"id":"1609.07593","kind":"arxiv","version":2}},"canonical_sha256":"438506b6a5aeca08eb47dad5be1a84eae8be91a9b67a3442cf577a1bb1203843","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"438506b6a5aeca08eb47dad5be1a84eae8be91a9b67a3442cf577a1bb1203843","first_computed_at":"2026-05-18T01:02:20.433569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:20.433569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JbBC5vxNdI8Z8YJ4HFOQE7nJfOPS9GsSSk6hNiRAVkbVawSQ67V/V+Mk++WNtAnTCz58MOwjfkbz9A8B/najCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:20.434230Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.07593","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf729f616d7ed11fec24f32cea96b206349cd497fc2eae3c473d983bd222ee62","sha256:f2f4a2222f9dd4d6e5b3108c1521797e35c9145ace68b39b964eaadefea0eb8a"],"state_sha256":"39c0c22197e6c1fc6b5bfdd031cb94002a134417b8abf71af62fe85d73e2a218"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7KnD4HHonyoedykLZ2JFSQ2/HvuqKbhDncLZkOE6+1RNCmgGjjmccUIBOD3Sb5F88ODfqpfufvuSsRp3QPN9CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T08:29:05.725742Z","bundle_sha256":"e9f8d760eaedc299c5724b1c6aa358544fedfe9be0e63da64f77ba60216ba354"}}