{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:LQUYTE4BWZNM52DKBYM3VAGIAH","short_pith_number":"pith:LQUYTE4B","canonical_record":{"source":{"id":"1009.3723","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-20T07:49:23Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"f1c2a511eea47fa65df62390ae35c7c63bc3c7163b25d18cef8dd3a07c2d2488","abstract_canon_sha256":"df4589bde0787e1c69a0c9eb6e0d1286d5d97144e1d0f3afdede48cf56fd4478"},"schema_version":"1.0"},"canonical_sha256":"5c29899381b65acee86a0e19ba80c801d9a5ed357583e6a08121ac593a28824d","source":{"kind":"arxiv","id":"1009.3723","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.3723","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"arxiv_version","alias_value":"1009.3723v3","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3723","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_12","alias_value":"LQUYTE4BWZNM","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"LQUYTE4BWZNM52DK","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"LQUYTE4B","created_at":"2026-07-05T00:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:LQUYTE4BWZNM52DKBYM3VAGIAH","target":"record","payload":{"canonical_record":{"source":{"id":"1009.3723","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-20T07:49:23Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"f1c2a511eea47fa65df62390ae35c7c63bc3c7163b25d18cef8dd3a07c2d2488","abstract_canon_sha256":"df4589bde0787e1c69a0c9eb6e0d1286d5d97144e1d0f3afdede48cf56fd4478"},"schema_version":"1.0"},"canonical_sha256":"5c29899381b65acee86a0e19ba80c801d9a5ed357583e6a08121ac593a28824d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:26:53.229965Z","signature_b64":"amkQNR0taBeRGKqmRk/zUEyqdEsoDDhSfAf6M5zEint+2GtmazyT3XCz4rBHNOU5rvFn9lbobSlKgu2YqQYOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c29899381b65acee86a0e19ba80c801d9a5ed357583e6a08121ac593a28824d","last_reissued_at":"2026-07-05T00:26:53.229566Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:26:53.229566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.3723","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:26:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rzhAOxLaXzGa3NskOz+4XzsGaYrKNymE855oX4KMXivZzE1Cwtkm9jycbyNzTgoTgb9SkZwk/u1Pv6AzmkyADQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T14:06:33.946967Z"},"content_sha256":"6cce6804852774546ef95b2e5e02c6bf990dcb8726ac4360afc0be0c2a73fd34","schema_version":"1.0","event_id":"sha256:6cce6804852774546ef95b2e5e02c6bf990dcb8726ac4360afc0be0c2a73fd34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:LQUYTE4BWZNM52DKBYM3VAGIAH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The probability of long cycles in interchange processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.PR","authors_text":"Gady Kozma, Gil Alon","submitted_at":"2010-09-20T07:49:23Z","abstract_excerpt":"We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3723","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1009.3723/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:26:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9LNjwYck1Y2tQqip+EewuFcoALWVFZ12rga4qYULwwiP+p3ls1KkTMOtL+fsDMucBGzHQtzjsQf6DzxxVuNpCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T14:06:33.947454Z"},"content_sha256":"59ab624f919788279b0070b920c73300cc57124936b4d4669e16a7573a8713cb","schema_version":"1.0","event_id":"sha256:59ab624f919788279b0070b920c73300cc57124936b4d4669e16a7573a8713cb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/bundle.json","state_url":"https://pith.science/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/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":"2026-08-21T14:06:33Z","links":{"resolver":"https://pith.science/pith/LQUYTE4BWZNM52DKBYM3VAGIAH","bundle":"https://pith.science/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/bundle.json","state":"https://pith.science/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LQUYTE4BWZNM52DKBYM3VAGIAH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:LQUYTE4BWZNM52DKBYM3VAGIAH","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":"df4589bde0787e1c69a0c9eb6e0d1286d5d97144e1d0f3afdede48cf56fd4478","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-20T07:49:23Z","title_canon_sha256":"f1c2a511eea47fa65df62390ae35c7c63bc3c7163b25d18cef8dd3a07c2d2488"},"schema_version":"1.0","source":{"id":"1009.3723","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.3723","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"arxiv_version","alias_value":"1009.3723v3","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3723","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_12","alias_value":"LQUYTE4BWZNM","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"LQUYTE4BWZNM52DK","created_at":"2026-07-05T00:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"LQUYTE4B","created_at":"2026-07-05T00:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:59ab624f919788279b0070b920c73300cc57124936b4d4669e16a7573a8713cb","target":"graph","created_at":"2026-07-05T00:26:53Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1009.3723/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.","authors_text":"Gady Kozma, Gil Alon","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-20T07:49:23Z","title":"The probability of long cycles in interchange processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3723","kind":"arxiv","version":3},"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":"sha256:6cce6804852774546ef95b2e5e02c6bf990dcb8726ac4360afc0be0c2a73fd34","target":"record","created_at":"2026-07-05T00:26:53Z","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":"df4589bde0787e1c69a0c9eb6e0d1286d5d97144e1d0f3afdede48cf56fd4478","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-20T07:49:23Z","title_canon_sha256":"f1c2a511eea47fa65df62390ae35c7c63bc3c7163b25d18cef8dd3a07c2d2488"},"schema_version":"1.0","source":{"id":"1009.3723","kind":"arxiv","version":3}},"canonical_sha256":"5c29899381b65acee86a0e19ba80c801d9a5ed357583e6a08121ac593a28824d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c29899381b65acee86a0e19ba80c801d9a5ed357583e6a08121ac593a28824d","first_computed_at":"2026-07-05T00:26:53.229566Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:26:53.229566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"amkQNR0taBeRGKqmRk/zUEyqdEsoDDhSfAf6M5zEint+2GtmazyT3XCz4rBHNOU5rvFn9lbobSlKgu2YqQYOCw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:26:53.229965Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.3723","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6cce6804852774546ef95b2e5e02c6bf990dcb8726ac4360afc0be0c2a73fd34","sha256:59ab624f919788279b0070b920c73300cc57124936b4d4669e16a7573a8713cb"],"state_sha256":"5704d2e166f095712965956be782ed34a4ba1e4a01ead59ed391991c4c2243a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BT7hRXZuLxO1BT49cPS4c0uXEAH0KL4ily0SUFfjq/zKPSlaL9UfVfZW6X6PEsbuW6c/N59b0e2aEDvg43yLAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T14:06:33.951321Z","bundle_sha256":"548b268aa7580c09efc317d792861376950843f3e54e73b8712ecdd5fc07941f"}}