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A representation taking (1+\\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant \\epsilon <= 1. A representation taking the optimal \\ceil{\\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time.\n  We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \\rightarrow [n] so "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.1983","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-08-09T17:01:12Z","cross_cats_sorted":[],"title_canon_sha256":"044f9b7512739148af0a4cc7a4ecf4406cb0c39c08ab5a65e3e156f6a4bfc884","abstract_canon_sha256":"7e3a3e7193c6072248dae58489c696e98e15465cb90c91b288015a0e4d0ee392"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:50.758496Z","signature_b64":"eCIKxouyfzYskqCJDCl4Ol/PIMUdeIz47mwc35DX1Qt435/5S3mzvPCC8BhQUR1oRQgWhXgBEjujm7lijZVrAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"891806aae7a75219a5b35c85ce812abb85c2fb472ec01985a1594d8f2f7e6a33","last_reissued_at":"2026-05-18T04:15:50.757905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:50.757905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Succinct Representations of Permutations and Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"J. 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A representation taking the optimal \\ceil{\\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time.\n  We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \\rightarrow [n] so "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.1983","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.1983","created_at":"2026-05-18T04:15:50.758012+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.1983v1","created_at":"2026-05-18T04:15:50.758012+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.1983","created_at":"2026-05-18T04:15:50.758012+00:00"},{"alias_kind":"pith_short_12","alias_value":"REMANKXHU5JB","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"REMANKXHU5JBTJNT","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"REMANKXH","created_at":"2026-05-18T12:26:41.206345+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO","json":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO.json","graph_json":"https://pith.science/api/pith-number/REMANKXHU5JBTJNTLSC45AJKXO/graph.json","events_json":"https://pith.science/api/pith-number/REMANKXHU5JBTJNTLSC45AJKXO/events.json","paper":"https://pith.science/paper/REMANKXH"},"agent_actions":{"view_html":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO","download_json":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO.json","view_paper":"https://pith.science/paper/REMANKXH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.1983&json=true","fetch_graph":"https://pith.science/api/pith-number/REMANKXHU5JBTJNTLSC45AJKXO/graph.json","fetch_events":"https://pith.science/api/pith-number/REMANKXHU5JBTJNTLSC45AJKXO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO/action/storage_attestation","attest_author":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO/action/author_attestation","sign_citation":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO/action/citation_signature","submit_replication":"https://pith.science/pith/REMANKXHU5JBTJNTLSC45AJKXO/action/replication_record"}},"created_at":"2026-05-18T04:15:50.758012+00:00","updated_at":"2026-05-18T04:15:50.758012+00:00"}