{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:TWYTAX3NR5UMJR2H6V5FHDYA26","short_pith_number":"pith:TWYTAX3N","canonical_record":{"source":{"id":"2408.01211","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-02T11:54:12Z","cross_cats_sorted":[],"title_canon_sha256":"c59f090d594616223b87cba131fce79771fe0bb560ecb572d6f6eb40e519d22e","abstract_canon_sha256":"1617af5fd078f2b1e5932b245273d910550ff6b0ab9188a295a14ea5773163c9"},"schema_version":"1.0"},"canonical_sha256":"9db1305f6d8f68c4c747f57a538f00d7bbb9e65b5404fae3082428958b7c350a","source":{"kind":"arxiv","id":"2408.01211","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.01211","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"arxiv_version","alias_value":"2408.01211v2","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.01211","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_12","alias_value":"TWYTAX3NR5UM","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_16","alias_value":"TWYTAX3NR5UMJR2H","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_8","alias_value":"TWYTAX3N","created_at":"2026-07-05T08:58:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:TWYTAX3NR5UMJR2H6V5FHDYA26","target":"record","payload":{"canonical_record":{"source":{"id":"2408.01211","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-02T11:54:12Z","cross_cats_sorted":[],"title_canon_sha256":"c59f090d594616223b87cba131fce79771fe0bb560ecb572d6f6eb40e519d22e","abstract_canon_sha256":"1617af5fd078f2b1e5932b245273d910550ff6b0ab9188a295a14ea5773163c9"},"schema_version":"1.0"},"canonical_sha256":"9db1305f6d8f68c4c747f57a538f00d7bbb9e65b5404fae3082428958b7c350a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:58:25.646073Z","signature_b64":"Xy/Iu3soriujQTbfJvws0u3UnRsxXFqfWnEymDvBcRZeUsIU9AUNrhMcXpItreeZNO5jduTK5kAZLpP/GCHXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9db1305f6d8f68c4c747f57a538f00d7bbb9e65b5404fae3082428958b7c350a","last_reissued_at":"2026-07-05T08:58:25.645581Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:58:25.645581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2408.01211","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":"2026-07-05T08:58:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UIAEtmhOHsQ3TD+hTVh7QwbAMf0IqntZrTkCnLOiJz4wABjoD2zx/q3Yc+OziL/pJW4MdHzaUaXUMgbomqsjAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T11:19:19.666792Z"},"content_sha256":"f464f0598f09c42129e15642eff531dcac6e05898173cc3aebbeee6b500a790a","schema_version":"1.0","event_id":"sha256:f464f0598f09c42129e15642eff531dcac6e05898173cc3aebbeee6b500a790a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:TWYTAX3NR5UMJR2H6V5FHDYA26","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Descents and inversions in powers of permutations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Yan, Stijn Cambie","submitted_at":"2024-08-02T11:54:12Z","abstract_excerpt":"In this paper, we generalise several recent results by Archer and Geary on descents in powers of permutations, and confirm all their conjectures. Specifically, for all $k\\in\\mathbb{Z}^+$, we prove explicit formulas for the expected numbers of descents and inversions in the $k$-th powers of permutations in $\\mathcal{S}_n$ for all $n\\geq2k+1$. We also compute the number of Grassmanian permutations in $\\mathcal{S}_n$ whose $k$-th powers remain Grassmanian, and the number of permutations in $\\mathcal{S}_n$ whose $k$-th powers have the maximum number of descents."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.01211","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.01211/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":"2026-07-05T08:58:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s+Mtq9tW63jVYJYfWdsvmcwkptj6MoAZazvj42kBQtP6lhYGuv3Qy32iiHZ8SvFTA/ubLfw3Wnj9ECyewCyHDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T11:19:19.667171Z"},"content_sha256":"b1bc40c822738e22df79d3bd648a4c6ac48a237ce47c9d0d8e58be8c17cb4c20","schema_version":"1.0","event_id":"sha256:b1bc40c822738e22df79d3bd648a4c6ac48a237ce47c9d0d8e58be8c17cb4c20"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/bundle.json","state_url":"https://pith.science/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/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":"2026-07-07T11:19:19Z","links":{"resolver":"https://pith.science/pith/TWYTAX3NR5UMJR2H6V5FHDYA26","bundle":"https://pith.science/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/bundle.json","state":"https://pith.science/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TWYTAX3NR5UMJR2H6V5FHDYA26/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:TWYTAX3NR5UMJR2H6V5FHDYA26","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":"1617af5fd078f2b1e5932b245273d910550ff6b0ab9188a295a14ea5773163c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-02T11:54:12Z","title_canon_sha256":"c59f090d594616223b87cba131fce79771fe0bb560ecb572d6f6eb40e519d22e"},"schema_version":"1.0","source":{"id":"2408.01211","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.01211","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"arxiv_version","alias_value":"2408.01211v2","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.01211","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_12","alias_value":"TWYTAX3NR5UM","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_16","alias_value":"TWYTAX3NR5UMJR2H","created_at":"2026-07-05T08:58:25Z"},{"alias_kind":"pith_short_8","alias_value":"TWYTAX3N","created_at":"2026-07-05T08:58:25Z"}],"graph_snapshots":[{"event_id":"sha256:b1bc40c822738e22df79d3bd648a4c6ac48a237ce47c9d0d8e58be8c17cb4c20","target":"graph","created_at":"2026-07-05T08:58:25Z","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/2408.01211/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we generalise several recent results by Archer and Geary on descents in powers of permutations, and confirm all their conjectures. Specifically, for all $k\\in\\mathbb{Z}^+$, we prove explicit formulas for the expected numbers of descents and inversions in the $k$-th powers of permutations in $\\mathcal{S}_n$ for all $n\\geq2k+1$. We also compute the number of Grassmanian permutations in $\\mathcal{S}_n$ whose $k$-th powers remain Grassmanian, and the number of permutations in $\\mathcal{S}_n$ whose $k$-th powers have the maximum number of descents.","authors_text":"Jun Yan, Stijn Cambie","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-02T11:54:12Z","title":"Descents and inversions in powers of permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.01211","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:f464f0598f09c42129e15642eff531dcac6e05898173cc3aebbeee6b500a790a","target":"record","created_at":"2026-07-05T08:58:25Z","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":"1617af5fd078f2b1e5932b245273d910550ff6b0ab9188a295a14ea5773163c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-02T11:54:12Z","title_canon_sha256":"c59f090d594616223b87cba131fce79771fe0bb560ecb572d6f6eb40e519d22e"},"schema_version":"1.0","source":{"id":"2408.01211","kind":"arxiv","version":2}},"canonical_sha256":"9db1305f6d8f68c4c747f57a538f00d7bbb9e65b5404fae3082428958b7c350a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9db1305f6d8f68c4c747f57a538f00d7bbb9e65b5404fae3082428958b7c350a","first_computed_at":"2026-07-05T08:58:25.645581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:58:25.645581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xy/Iu3soriujQTbfJvws0u3UnRsxXFqfWnEymDvBcRZeUsIU9AUNrhMcXpItreeZNO5jduTK5kAZLpP/GCHXCw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:58:25.646073Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.01211","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f464f0598f09c42129e15642eff531dcac6e05898173cc3aebbeee6b500a790a","sha256:b1bc40c822738e22df79d3bd648a4c6ac48a237ce47c9d0d8e58be8c17cb4c20"],"state_sha256":"b075731190bc78e9a983f744cf38cbbd770272535b0a84b0396847a91809d22c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hKkqMsuugdTrFJ+JrNzjqY+/j2WPGxNpkJskEcuXisjX3BS99Q3HfmJkubMJ1rlcQ2c1wScmT1UB8JIrOYk9Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T11:19:19.669038Z","bundle_sha256":"b3af2fa5e41a93a46a1b3e9a25993687e42c665b72364518a2aa4e506bf9e27a"}}