{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:L6N2VLZPQODV6ZU2MZMAABZRDK","short_pith_number":"pith:L6N2VLZP","canonical_record":{"source":{"id":"1309.7117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-27T05:28:43Z","cross_cats_sorted":[],"title_canon_sha256":"e0e91481eba1531a90018590316b4297b3084e9623745e701080dc4360e01824","abstract_canon_sha256":"eb2f2526b421ee4ea2697475da59f5ad0f2ed281505473e817a58edc3788b5c3"},"schema_version":"1.0"},"canonical_sha256":"5f9baaaf2f83875f669a66580007311a8e2319068003263d01b72fe6a3276b47","source":{"kind":"arxiv","id":"1309.7117","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7117","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7117v1","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7117","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"pith_short_12","alias_value":"L6N2VLZPQODV","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"L6N2VLZPQODV6ZU2","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"L6N2VLZP","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:L6N2VLZPQODV6ZU2MZMAABZRDK","target":"record","payload":{"canonical_record":{"source":{"id":"1309.7117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-27T05:28:43Z","cross_cats_sorted":[],"title_canon_sha256":"e0e91481eba1531a90018590316b4297b3084e9623745e701080dc4360e01824","abstract_canon_sha256":"eb2f2526b421ee4ea2697475da59f5ad0f2ed281505473e817a58edc3788b5c3"},"schema_version":"1.0"},"canonical_sha256":"5f9baaaf2f83875f669a66580007311a8e2319068003263d01b72fe6a3276b47","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:00.198547Z","signature_b64":"BQOk1NCYWHPzn5Gx6nIhqwi68JQ86TgtwFwiScxXWhET6kLdoLXKie3/cV1u7kDPMzK0o2aWIhLBlOWu4lZqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f9baaaf2f83875f669a66580007311a8e2319068003263d01b72fe6a3276b47","last_reissued_at":"2026-05-18T03:12:00.197892Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:00.197892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.7117","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":"2026-05-18T03:12:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A8/W9JmmaOHQZawzUfzSWxM6MyO0hkIzAnFtpKkAbcvTOsOlZVCfEqR+mpVFYBHCXiO6ebD4/J/vZKYl11ZQCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:39:43.348113Z"},"content_sha256":"be66bd728bbcd76881d2fcec7cf99972043c082b8a12c58dda48a4eb10b23104","schema_version":"1.0","event_id":"sha256:be66bd728bbcd76881d2fcec7cf99972043c082b8a12c58dda48a4eb10b23104"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:L6N2VLZPQODV6ZU2MZMAABZRDK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Using functional equations to enumerate 1324-avoiding permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brian Nakamura, Fredrik Johansson","submitted_at":"2013-09-27T05:28:43Z","abstract_excerpt":"We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n=31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7117","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":"2026-05-18T03:12:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MadPSaZaB7IGosLylXIOLV4xRj+G0Jg7e5V5OVdAsLx9BP4h/KozItqDh3TgctlqNwiADeZtnUSQ3IOwJmAMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:39:43.348715Z"},"content_sha256":"da24ab98b24125f6317e33b8e7314d52bc8d157d5c2dee4eb29adf41db2d88de","schema_version":"1.0","event_id":"sha256:da24ab98b24125f6317e33b8e7314d52bc8d157d5c2dee4eb29adf41db2d88de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/bundle.json","state_url":"https://pith.science/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/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":"2026-05-26T12:39:43Z","links":{"resolver":"https://pith.science/pith/L6N2VLZPQODV6ZU2MZMAABZRDK","bundle":"https://pith.science/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/bundle.json","state":"https://pith.science/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L6N2VLZPQODV6ZU2MZMAABZRDK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:L6N2VLZPQODV6ZU2MZMAABZRDK","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":"eb2f2526b421ee4ea2697475da59f5ad0f2ed281505473e817a58edc3788b5c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-27T05:28:43Z","title_canon_sha256":"e0e91481eba1531a90018590316b4297b3084e9623745e701080dc4360e01824"},"schema_version":"1.0","source":{"id":"1309.7117","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7117","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7117v1","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7117","created_at":"2026-05-18T03:12:00Z"},{"alias_kind":"pith_short_12","alias_value":"L6N2VLZPQODV","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"L6N2VLZPQODV6ZU2","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"L6N2VLZP","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:da24ab98b24125f6317e33b8e7314d52bc8d157d5c2dee4eb29adf41db2d88de","target":"graph","created_at":"2026-05-18T03:12:00Z","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 the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n=31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.","authors_text":"Brian Nakamura, Fredrik Johansson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-27T05:28:43Z","title":"Using functional equations to enumerate 1324-avoiding permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7117","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:be66bd728bbcd76881d2fcec7cf99972043c082b8a12c58dda48a4eb10b23104","target":"record","created_at":"2026-05-18T03:12:00Z","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":"eb2f2526b421ee4ea2697475da59f5ad0f2ed281505473e817a58edc3788b5c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-27T05:28:43Z","title_canon_sha256":"e0e91481eba1531a90018590316b4297b3084e9623745e701080dc4360e01824"},"schema_version":"1.0","source":{"id":"1309.7117","kind":"arxiv","version":1}},"canonical_sha256":"5f9baaaf2f83875f669a66580007311a8e2319068003263d01b72fe6a3276b47","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f9baaaf2f83875f669a66580007311a8e2319068003263d01b72fe6a3276b47","first_computed_at":"2026-05-18T03:12:00.197892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:00.197892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BQOk1NCYWHPzn5Gx6nIhqwi68JQ86TgtwFwiScxXWhET6kLdoLXKie3/cV1u7kDPMzK0o2aWIhLBlOWu4lZqAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:00.198547Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7117","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be66bd728bbcd76881d2fcec7cf99972043c082b8a12c58dda48a4eb10b23104","sha256:da24ab98b24125f6317e33b8e7314d52bc8d157d5c2dee4eb29adf41db2d88de"],"state_sha256":"bfa7c020dbf3e10331e1913284420b840765a490d9d31ddfd734d0309a1cd92d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+ZrNnfjpdTF1BcNdr+3geg1Pl46K1TjQ9ut3McqJNjVAcSDL51qcpKHdWz8aXdOwlJHNmPrDy1Bd0JHUE+LVDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T12:39:43.352200Z","bundle_sha256":"f01241da3f6995db0b27d24ad4bdac6583ecbd78fb7fbc5a3c8f27dadb004166"}}