{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:WAEFSIXKOKS3ZATIK3WRTFF4DJ","short_pith_number":"pith:WAEFSIXK","canonical_record":{"source":{"id":"1106.5781","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-27T23:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"b4f5476c1396e5b2682e694bd3ea0e47cb3c9816b1945931e7982751d241fd32","abstract_canon_sha256":"b770a092db695a9d2d5d8bc9a0e13660261355a61d184307852f6f7714159276"},"schema_version":"1.0"},"canonical_sha256":"b0085922ea72a5bc826856ed1994bc1a7957c558fa005ff06ce51bd64a46d1e5","source":{"kind":"arxiv","id":"1106.5781","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5781","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5781v2","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5781","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"WAEFSIXKOKS3","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WAEFSIXKOKS3ZATI","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WAEFSIXK","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:WAEFSIXKOKS3ZATIK3WRTFF4DJ","target":"record","payload":{"canonical_record":{"source":{"id":"1106.5781","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-27T23:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"b4f5476c1396e5b2682e694bd3ea0e47cb3c9816b1945931e7982751d241fd32","abstract_canon_sha256":"b770a092db695a9d2d5d8bc9a0e13660261355a61d184307852f6f7714159276"},"schema_version":"1.0"},"canonical_sha256":"b0085922ea72a5bc826856ed1994bc1a7957c558fa005ff06ce51bd64a46d1e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:43.023709Z","signature_b64":"IsVq58d1hjZurCpCPHpxjp4ZnuZfLY406SIq0JlZU+FpawSGVhKBR6v9oABl1qMU8pF3aTjRm8aju/21796zAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0085922ea72a5bc826856ed1994bc1a7957c558fa005ff06ce51bd64a46d1e5","last_reissued_at":"2026-05-18T04:07:43.023320Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:43.023320Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.5781","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":"2026-05-18T04:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a207OjTqXQM+cpAjMPVBQOQ3bRr11yM1YHL6hka46e2vVs9yg4ryM6/+r6gunOeXKuvZz3sCmc4GKMKgzRBIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:15:29.022601Z"},"content_sha256":"408acbd4e60b8e5bea7ca1e6c01a65f4a5d9f6fe69f8387b00309885dc1ee057","schema_version":"1.0","event_id":"sha256:408acbd4e60b8e5bea7ca1e6c01a65f4a5d9f6fe69f8387b00309885dc1ee057"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:WAEFSIXKOKS3ZATIK3WRTFF4DJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Derivative polynomials and permutations by numbers of interior peaks and left peaks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shi-Mei Ma","submitted_at":"2011-06-27T23:19:55Z","abstract_excerpt":"Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left peaks over the symmetric group. Properties of the generating functions for the numbers of interior and left peaks over the symmetric group, including recurrence relations, generating functions and real-rootedness, are studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5781","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":"2026-05-18T04:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FLzmEi3OD6ug2I+X4sXcsfvRYO8/xk3xnuW6AnuiOW3th5l9h9xrm3dkVxIwyAaRayC4RJ/2JJ23ABbEYPF2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:15:29.022948Z"},"content_sha256":"373f80588fdb2d5012cb0b565f927e1ccdbf35bc90e4ddd82b4707564b7ad662","schema_version":"1.0","event_id":"sha256:373f80588fdb2d5012cb0b565f927e1ccdbf35bc90e4ddd82b4707564b7ad662"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/bundle.json","state_url":"https://pith.science/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/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":"2026-06-01T16:15:29Z","links":{"resolver":"https://pith.science/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ","bundle":"https://pith.science/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/bundle.json","state":"https://pith.science/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WAEFSIXKOKS3ZATIK3WRTFF4DJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:WAEFSIXKOKS3ZATIK3WRTFF4DJ","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":"b770a092db695a9d2d5d8bc9a0e13660261355a61d184307852f6f7714159276","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-27T23:19:55Z","title_canon_sha256":"b4f5476c1396e5b2682e694bd3ea0e47cb3c9816b1945931e7982751d241fd32"},"schema_version":"1.0","source":{"id":"1106.5781","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5781","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5781v2","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5781","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"WAEFSIXKOKS3","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WAEFSIXKOKS3ZATI","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WAEFSIXK","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:373f80588fdb2d5012cb0b565f927e1ccdbf35bc90e4ddd82b4707564b7ad662","target":"graph","created_at":"2026-05-18T04:07:43Z","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":"Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left peaks over the symmetric group. Properties of the generating functions for the numbers of interior and left peaks over the symmetric group, including recurrence relations, generating functions and real-rootedness, are studied.","authors_text":"Shi-Mei Ma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-27T23:19:55Z","title":"Derivative polynomials and permutations by numbers of interior peaks and left peaks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5781","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":"sha256:408acbd4e60b8e5bea7ca1e6c01a65f4a5d9f6fe69f8387b00309885dc1ee057","target":"record","created_at":"2026-05-18T04:07:43Z","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":"b770a092db695a9d2d5d8bc9a0e13660261355a61d184307852f6f7714159276","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-27T23:19:55Z","title_canon_sha256":"b4f5476c1396e5b2682e694bd3ea0e47cb3c9816b1945931e7982751d241fd32"},"schema_version":"1.0","source":{"id":"1106.5781","kind":"arxiv","version":2}},"canonical_sha256":"b0085922ea72a5bc826856ed1994bc1a7957c558fa005ff06ce51bd64a46d1e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0085922ea72a5bc826856ed1994bc1a7957c558fa005ff06ce51bd64a46d1e5","first_computed_at":"2026-05-18T04:07:43.023320Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:43.023320Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IsVq58d1hjZurCpCPHpxjp4ZnuZfLY406SIq0JlZU+FpawSGVhKBR6v9oABl1qMU8pF3aTjRm8aju/21796zAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:43.023709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.5781","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:408acbd4e60b8e5bea7ca1e6c01a65f4a5d9f6fe69f8387b00309885dc1ee057","sha256:373f80588fdb2d5012cb0b565f927e1ccdbf35bc90e4ddd82b4707564b7ad662"],"state_sha256":"1e921f15804decd152da5fdbcde13f0f41f69f9e92f2a44963b3f8374ef3916b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FLimfjcuG/uJAraIwkLcwd+gStBR/iQwiYyFD1u6GSZEr8aKn95Zhwn+O19ULt7F2hKg9ZQc0ptXsYJ7s2ILDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T16:15:29.024900Z","bundle_sha256":"152bca2d19eea98b35ac19251a155d0f2ae28645514763fa5982e369bcff3b26"}}