{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FMHOGJCJTTWM7PIZZGT4TBDKT6","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":"c4c9d7f02747da59b756d0f6cca3d08ddd349d889a1566c7e5d8c5f2185d53e0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T16:03:42Z","title_canon_sha256":"51aceeacebd88440df31be11d0df05a734996285ed139f792ade753e930c0ff8"},"schema_version":"1.0","source":{"id":"2605.27214","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.27214","created_at":"2026-05-27T02:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"2605.27214v1","created_at":"2026-05-27T02:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27214","created_at":"2026-05-27T02:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"FMHOGJCJTTWM","created_at":"2026-05-27T02:05:48Z"},{"alias_kind":"pith_short_16","alias_value":"FMHOGJCJTTWM7PIZ","created_at":"2026-05-27T02:05:48Z"},{"alias_kind":"pith_short_8","alias_value":"FMHOGJCJ","created_at":"2026-05-27T02:05:48Z"}],"graph_snapshots":[{"event_id":"sha256:6336b0b053a924654a9c527cb2b8ed80230cc15a53e6ac0965beb27fb0a22b6f","target":"graph","created_at":"2026-05-27T02:05:48Z","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/2605.27214/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Problem 8.1 in Astaiza et. al. asks about the relationship between the cycle decomposition of a permutation $\\sigma$ and that of its symmetric tensor power $\\sigma ^{\\odot k}$. In this paper, we investigate this question and give formulas for computing the number of fixed points and, in the case of a permutation containing at most one cycle of length greater than one, the number of $s$-cycles.","authors_text":"Diego Villamizar, Sebastian Caballero","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T16:03:42Z","title":"On the cycle structure of the symmetric tensor power of permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27214","kind":"arxiv","version":1},"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":"sha256:5e0f758d66b925fc52d3771aae66df12a0933e617aefe2af11efba3a779e90d1","target":"record","created_at":"2026-05-27T02:05:48Z","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":"c4c9d7f02747da59b756d0f6cca3d08ddd349d889a1566c7e5d8c5f2185d53e0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T16:03:42Z","title_canon_sha256":"51aceeacebd88440df31be11d0df05a734996285ed139f792ade753e930c0ff8"},"schema_version":"1.0","source":{"id":"2605.27214","kind":"arxiv","version":1}},"canonical_sha256":"2b0ee324499ceccfbd19c9a7c9846a9f8b1dfd1a9606634c2bd979b793b53e8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b0ee324499ceccfbd19c9a7c9846a9f8b1dfd1a9606634c2bd979b793b53e8e","first_computed_at":"2026-05-27T02:05:48.408397Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-27T02:05:48.408397Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x2MkIbUq8LZ2IQjbjTgfzrRmjPrzrurzfrmSC4FPV6juR4bug2vG4dBFWXPtRxc4raIlX2uvCq2sBqQLZ3PiDw==","signature_status":"signed_v1","signed_at":"2026-05-27T02:05:48.409154Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.27214","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e0f758d66b925fc52d3771aae66df12a0933e617aefe2af11efba3a779e90d1","sha256:6336b0b053a924654a9c527cb2b8ed80230cc15a53e6ac0965beb27fb0a22b6f"],"state_sha256":"73510e639dabbb604a3f68e1884208020e6e34ba84ebf532e63c0f71ef69c4f9"}