{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6DJVNJWC4SUUIH2GKHOWRVUPMH","short_pith_number":"pith:6DJVNJWC","canonical_record":{"source":{"id":"1807.10450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","cross_cats_sorted":[],"title_canon_sha256":"1da7b2057c7273f4e626fc3d466f2ca66636cb921d8c144405e83adf6856ebd6","abstract_canon_sha256":"05164db87275bd082c66c356a08fd0f2115d83166062f9986101297f7554093f"},"schema_version":"1.0"},"canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","source":{"kind":"arxiv","id":"1807.10450","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10450","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10450v2","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10450","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"6DJVNJWC4SUU","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6DJVNJWC4SUUIH2G","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6DJVNJWC","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6DJVNJWC4SUUIH2GKHOWRVUPMH","target":"record","payload":{"canonical_record":{"source":{"id":"1807.10450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","cross_cats_sorted":[],"title_canon_sha256":"1da7b2057c7273f4e626fc3d466f2ca66636cb921d8c144405e83adf6856ebd6","abstract_canon_sha256":"05164db87275bd082c66c356a08fd0f2115d83166062f9986101297f7554093f"},"schema_version":"1.0"},"canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:17.231943Z","signature_b64":"ocHPvwBaY3Kt7SGBjsqt2xStigomu8O/4ORvJ2RdQ6wWzP42LFU589jvLkwXXrx6gmQSKTRcEOlQXbZSTWPFDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","last_reissued_at":"2026-05-17T23:48:17.231252Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:17.231252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.10450","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":"2026-05-17T23:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AzmRteSje8Gykmpc+NYsluIDroTLDGSeK06ViuuJNCY3anspLdQDFmHOAMlNS0XQnbcn5gUQA8gHT+ipEyThAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T01:13:22.427301Z"},"content_sha256":"7d111069221e55db379fc91dff4b4cd9aacf24bd0cfe07d3667d66b33f2e307b","schema_version":"1.0","event_id":"sha256:7d111069221e55db379fc91dff4b4cd9aacf24bd0cfe07d3667d66b33f2e307b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6DJVNJWC4SUUIH2GKHOWRVUPMH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Permutations with orders coprime to a given integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheryl E. Praeger, John Bamberg, Scott Harper, S. P. Glasby","submitted_at":"2018-07-27T06:14:47Z","abstract_excerpt":"Let $m$ be a positive integer and let $\\rho(m,n)$ be the proportion of permutations of the symmetric group ${\\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\\rho(n,m)n^{1-\\frac{\\phi(m)}{m}}\\sim \\kappa_m$ where $\\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \\geqslant m$, \\[C(m) \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1} \\leqslant \\rho(n,m) \\leqslant \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1}\\] where $\\phi$ is Euler's totient function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10450","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":"2026-05-17T23:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jPgPcQFGBEVNBcEg8EhFJCy4OwrbbxNhWy8y7giMMy9KpHZ/YpzfLE6BLZzvjzAh8JRcEvHVAcrq/z1aUU4eAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T01:13:22.427663Z"},"content_sha256":"44de98b819e9988e400353121bb77a6d91f139b9ee8fbe9ef20a46479d633b24","schema_version":"1.0","event_id":"sha256:44de98b819e9988e400353121bb77a6d91f139b9ee8fbe9ef20a46479d633b24"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/bundle.json","state_url":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/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":"2026-06-03T01:13:22Z","links":{"resolver":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH","bundle":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/bundle.json","state":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6DJVNJWC4SUUIH2GKHOWRVUPMH","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":"05164db87275bd082c66c356a08fd0f2115d83166062f9986101297f7554093f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","title_canon_sha256":"1da7b2057c7273f4e626fc3d466f2ca66636cb921d8c144405e83adf6856ebd6"},"schema_version":"1.0","source":{"id":"1807.10450","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10450","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10450v2","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10450","created_at":"2026-05-17T23:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"6DJVNJWC4SUU","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6DJVNJWC4SUUIH2G","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6DJVNJWC","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:44de98b819e9988e400353121bb77a6d91f139b9ee8fbe9ef20a46479d633b24","target":"graph","created_at":"2026-05-17T23:48:17Z","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 $m$ be a positive integer and let $\\rho(m,n)$ be the proportion of permutations of the symmetric group ${\\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\\rho(n,m)n^{1-\\frac{\\phi(m)}{m}}\\sim \\kappa_m$ where $\\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \\geqslant m$, \\[C(m) \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1} \\leqslant \\rho(n,m) \\leqslant \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1}\\] where $\\phi$ is Euler's totient function.","authors_text":"Cheryl E. Praeger, John Bamberg, Scott Harper, S. P. Glasby","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","title":"Permutations with orders coprime to a given integer"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10450","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":"sha256:7d111069221e55db379fc91dff4b4cd9aacf24bd0cfe07d3667d66b33f2e307b","target":"record","created_at":"2026-05-17T23:48:17Z","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":"05164db87275bd082c66c356a08fd0f2115d83166062f9986101297f7554093f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","title_canon_sha256":"1da7b2057c7273f4e626fc3d466f2ca66636cb921d8c144405e83adf6856ebd6"},"schema_version":"1.0","source":{"id":"1807.10450","kind":"arxiv","version":2}},"canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","first_computed_at":"2026-05-17T23:48:17.231252Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:17.231252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ocHPvwBaY3Kt7SGBjsqt2xStigomu8O/4ORvJ2RdQ6wWzP42LFU589jvLkwXXrx6gmQSKTRcEOlQXbZSTWPFDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:17.231943Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.10450","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d111069221e55db379fc91dff4b4cd9aacf24bd0cfe07d3667d66b33f2e307b","sha256:44de98b819e9988e400353121bb77a6d91f139b9ee8fbe9ef20a46479d633b24"],"state_sha256":"5787ec625198a60a5a2c729a9c750ebde5cf9f59d871865b6aee0e19110b5cc2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"295IoEgrpD5OSMWLZw91Qrpy9cfQsbZmzrj51fQVnD5/5dHJM0HFqE5kAg3iUtLfmKHhudu7VVCMC6Ty0DYZDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T01:13:22.429605Z","bundle_sha256":"7e45af85d85661607f1183add226ac8219bad346218bdf2178d4f422f1107f55"}}