{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:FD6WHMG636I2ZX6DA5Z7GMM3EA","short_pith_number":"pith:FD6WHMG6","canonical_record":{"source":{"id":"1110.5014","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-23T03:26:08Z","cross_cats_sorted":[],"title_canon_sha256":"d373cdee6d59537f1196bd5f6b648e6ef7a38652c85c1ac63ca4412d5d7289a5","abstract_canon_sha256":"a321b8bfe94e4fda21ed40f883bb994f72e3528191c3325daf8b3049e1191240"},"schema_version":"1.0"},"canonical_sha256":"28fd63b0dedf91acdfc30773f3319b2021926e3afc2619475c7a506faf4cabd9","source":{"kind":"arxiv","id":"1110.5014","version":7},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.5014","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1110.5014v7","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5014","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"FD6WHMG636I2","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FD6WHMG636I2ZX6D","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FD6WHMG6","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:FD6WHMG636I2ZX6DA5Z7GMM3EA","target":"record","payload":{"canonical_record":{"source":{"id":"1110.5014","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-23T03:26:08Z","cross_cats_sorted":[],"title_canon_sha256":"d373cdee6d59537f1196bd5f6b648e6ef7a38652c85c1ac63ca4412d5d7289a5","abstract_canon_sha256":"a321b8bfe94e4fda21ed40f883bb994f72e3528191c3325daf8b3049e1191240"},"schema_version":"1.0"},"canonical_sha256":"28fd63b0dedf91acdfc30773f3319b2021926e3afc2619475c7a506faf4cabd9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:51.123410Z","signature_b64":"OMn2TG/X5sM3AfxnZh66tCtZf8dR1VHHynzynCGoH0eBHVC9L0+AgbUGIy9j94Xe9e3qLgN5gftBSBykKQyGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28fd63b0dedf91acdfc30773f3319b2021926e3afc2619475c7a506faf4cabd9","last_reissued_at":"2026-05-18T03:39:51.122957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:51.122957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.5014","source_version":7,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7kxB9p9uHHedNgDsx1X6xTJnf/cl+XEN5PdhOUIZMZrHoud2VOP/dxvcpo7dgTl4at9VNQFX9gKYOD8lXxvSDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:09:53.786869Z"},"content_sha256":"430a96574f7659f0b35376be447cfe8d2bcc493afb1912045d572eb13d0746cc","schema_version":"1.0","event_id":"sha256:430a96574f7659f0b35376be447cfe8d2bcc493afb1912045d572eb13d0746cc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:FD6WHMG636I2ZX6DA5Z7GMM3EA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Enumeration of permutations by number of alternating runs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shi-Mei Ma","submitted_at":"2011-10-23T03:26:08Z","abstract_excerpt":"Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers R(n,k). Moreover, we establish a connection between alternating runs and Andre permutations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5014","kind":"arxiv","version":7},"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":"2026-05-18T03:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2k+xv1oCm/HtVE720Uz2OvmxNUNwzkR2CbWu7qsJNXHdiWJk6bLVqm7JzDMt24UtjQSzN1dkTqjQ1JaPseu1Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:09:53.787217Z"},"content_sha256":"d34c47194ce6c5b46b331c7d2acbd2ba3fe5b69aef28dc6b42cb04acee729e74","schema_version":"1.0","event_id":"sha256:d34c47194ce6c5b46b331c7d2acbd2ba3fe5b69aef28dc6b42cb04acee729e74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/bundle.json","state_url":"https://pith.science/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/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":"2026-06-01T19:09:53Z","links":{"resolver":"https://pith.science/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA","bundle":"https://pith.science/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/bundle.json","state":"https://pith.science/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FD6WHMG636I2ZX6DA5Z7GMM3EA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:FD6WHMG636I2ZX6DA5Z7GMM3EA","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":"a321b8bfe94e4fda21ed40f883bb994f72e3528191c3325daf8b3049e1191240","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-23T03:26:08Z","title_canon_sha256":"d373cdee6d59537f1196bd5f6b648e6ef7a38652c85c1ac63ca4412d5d7289a5"},"schema_version":"1.0","source":{"id":"1110.5014","kind":"arxiv","version":7}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.5014","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1110.5014v7","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5014","created_at":"2026-05-18T03:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"FD6WHMG636I2","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FD6WHMG636I2ZX6D","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FD6WHMG6","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:d34c47194ce6c5b46b331c7d2acbd2ba3fe5b69aef28dc6b42cb04acee729e74","target":"graph","created_at":"2026-05-18T03:39:51Z","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":"Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers R(n,k). Moreover, we establish a connection between alternating runs and Andre permutations.","authors_text":"Shi-Mei Ma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-23T03:26:08Z","title":"Enumeration of permutations by number of alternating runs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5014","kind":"arxiv","version":7},"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":"sha256:430a96574f7659f0b35376be447cfe8d2bcc493afb1912045d572eb13d0746cc","target":"record","created_at":"2026-05-18T03:39:51Z","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":"a321b8bfe94e4fda21ed40f883bb994f72e3528191c3325daf8b3049e1191240","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-23T03:26:08Z","title_canon_sha256":"d373cdee6d59537f1196bd5f6b648e6ef7a38652c85c1ac63ca4412d5d7289a5"},"schema_version":"1.0","source":{"id":"1110.5014","kind":"arxiv","version":7}},"canonical_sha256":"28fd63b0dedf91acdfc30773f3319b2021926e3afc2619475c7a506faf4cabd9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28fd63b0dedf91acdfc30773f3319b2021926e3afc2619475c7a506faf4cabd9","first_computed_at":"2026-05-18T03:39:51.122957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:51.122957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OMn2TG/X5sM3AfxnZh66tCtZf8dR1VHHynzynCGoH0eBHVC9L0+AgbUGIy9j94Xe9e3qLgN5gftBSBykKQyGCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:51.123410Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.5014","source_kind":"arxiv","source_version":7}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:430a96574f7659f0b35376be447cfe8d2bcc493afb1912045d572eb13d0746cc","sha256:d34c47194ce6c5b46b331c7d2acbd2ba3fe5b69aef28dc6b42cb04acee729e74"],"state_sha256":"a1e1dee00918b7f0a08aade451645f327d730eeee01ed8dea68007a77b85595a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WZlSq6dA0YlMa+e0ev3o+wOaiRQs9ywLdDRha05t/lvpHKjbLImZUcLITtYglHrRgbf4gDAC9f7gHN/bwfbmCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T19:09:53.789080Z","bundle_sha256":"d9d136aa9bae7f9928efb3e0bed847dd1764e4881f76bdaceb63681c4e975769"}}