{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:R7ZBOX4CNYCIXIPPDETAR4T4PI","short_pith_number":"pith:R7ZBOX4C","canonical_record":{"source":{"id":"1402.3970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-17T11:36:19Z","cross_cats_sorted":[],"title_canon_sha256":"76427afd01c1118f09ea2e25fec16ffedc407fdbe2ea7afe642aa819fb6f1929","abstract_canon_sha256":"d6c4ccfe29b21d3f93d3df6f27a8a3c05d63a170f88888c5e40ffb5412ac71db"},"schema_version":"1.0"},"canonical_sha256":"8ff2175f826e048ba1ef192608f27c7a10ebc61348b6efca50e5dd8821bc0a9b","source":{"kind":"arxiv","id":"1402.3970","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.3970","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"arxiv_version","alias_value":"1402.3970v1","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.3970","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"pith_short_12","alias_value":"R7ZBOX4CNYCI","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R7ZBOX4CNYCIXIPP","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R7ZBOX4C","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:R7ZBOX4CNYCIXIPPDETAR4T4PI","target":"record","payload":{"canonical_record":{"source":{"id":"1402.3970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-17T11:36:19Z","cross_cats_sorted":[],"title_canon_sha256":"76427afd01c1118f09ea2e25fec16ffedc407fdbe2ea7afe642aa819fb6f1929","abstract_canon_sha256":"d6c4ccfe29b21d3f93d3df6f27a8a3c05d63a170f88888c5e40ffb5412ac71db"},"schema_version":"1.0"},"canonical_sha256":"8ff2175f826e048ba1ef192608f27c7a10ebc61348b6efca50e5dd8821bc0a9b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:54.987579Z","signature_b64":"np5eJDd5Z57/7hje/aFQtwiy/qUWNzpySfMqoBRiZLAukGBR8r3T6iJmPJbvxZtDgn1g9d455aPYv28WE0RBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ff2175f826e048ba1ef192608f27c7a10ebc61348b6efca50e5dd8821bc0a9b","last_reissued_at":"2026-05-18T02:58:54.987046Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:54.987046Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.3970","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fw1lU7385kErd/EQunEY996P4PR5bG4r2KzmDZaTomV7KiySifhljClGHskbU2C6YMX0LtdK9UCOP9L3pwHEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:31:09.421657Z"},"content_sha256":"7c0422b248cb10837221f4245461cc43d120a108452a07b011d8fd4787d5ba69","schema_version":"1.0","event_id":"sha256:7c0422b248cb10837221f4245461cc43d120a108452a07b011d8fd4787d5ba69"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:R7ZBOX4CNYCIXIPPDETAR4T4PI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Additive Combinatorics of Permutations of \\mathbb{Z}_n","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Deepak Rajendraprasad, L. Sunil Chandran, Nitin Singh","submitted_at":"2014-02-17T11:36:19Z","abstract_excerpt":"Let $\\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.3970","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xbHX44L6BuVCI/+EDcuDoWSNRgFNiRre5sDMicLDiF38beo2auBH3BYKK1tq30WYLvledYSl3BC3kYvz9GDTBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:31:09.421996Z"},"content_sha256":"235de4fc1025d9b9358c90ac9ea436cd88949717a77c19787239b2fd50e557a6","schema_version":"1.0","event_id":"sha256:235de4fc1025d9b9358c90ac9ea436cd88949717a77c19787239b2fd50e557a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/bundle.json","state_url":"https://pith.science/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/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":"2026-06-01T06:31:09Z","links":{"resolver":"https://pith.science/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI","bundle":"https://pith.science/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/bundle.json","state":"https://pith.science/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R7ZBOX4CNYCIXIPPDETAR4T4PI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:R7ZBOX4CNYCIXIPPDETAR4T4PI","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":"d6c4ccfe29b21d3f93d3df6f27a8a3c05d63a170f88888c5e40ffb5412ac71db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-17T11:36:19Z","title_canon_sha256":"76427afd01c1118f09ea2e25fec16ffedc407fdbe2ea7afe642aa819fb6f1929"},"schema_version":"1.0","source":{"id":"1402.3970","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.3970","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"arxiv_version","alias_value":"1402.3970v1","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.3970","created_at":"2026-05-18T02:58:54Z"},{"alias_kind":"pith_short_12","alias_value":"R7ZBOX4CNYCI","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R7ZBOX4CNYCIXIPP","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R7ZBOX4C","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:235de4fc1025d9b9358c90ac9ea436cd88949717a77c19787239b2fd50e557a6","target":"graph","created_at":"2026-05-18T02:58:54Z","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 $\\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we pro","authors_text":"Deepak Rajendraprasad, L. Sunil Chandran, Nitin Singh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-17T11:36:19Z","title":"On Additive Combinatorics of Permutations of \\mathbb{Z}_n"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.3970","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":"sha256:7c0422b248cb10837221f4245461cc43d120a108452a07b011d8fd4787d5ba69","target":"record","created_at":"2026-05-18T02:58:54Z","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":"d6c4ccfe29b21d3f93d3df6f27a8a3c05d63a170f88888c5e40ffb5412ac71db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-17T11:36:19Z","title_canon_sha256":"76427afd01c1118f09ea2e25fec16ffedc407fdbe2ea7afe642aa819fb6f1929"},"schema_version":"1.0","source":{"id":"1402.3970","kind":"arxiv","version":1}},"canonical_sha256":"8ff2175f826e048ba1ef192608f27c7a10ebc61348b6efca50e5dd8821bc0a9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ff2175f826e048ba1ef192608f27c7a10ebc61348b6efca50e5dd8821bc0a9b","first_computed_at":"2026-05-18T02:58:54.987046Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:54.987046Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"np5eJDd5Z57/7hje/aFQtwiy/qUWNzpySfMqoBRiZLAukGBR8r3T6iJmPJbvxZtDgn1g9d455aPYv28WE0RBCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:54.987579Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.3970","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c0422b248cb10837221f4245461cc43d120a108452a07b011d8fd4787d5ba69","sha256:235de4fc1025d9b9358c90ac9ea436cd88949717a77c19787239b2fd50e557a6"],"state_sha256":"f78da110c0aea2dad848e7471fcd93721be75414c3fe2dffc777e546393214d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BY8T+kaHGISO1vcI/8dPZlsfxS9WxVcXW2MScemZM0qZtF0sYcUDnDOzdCkNkYrOBFy6piDP++2zt4+4oRoKBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:31:09.424109Z","bundle_sha256":"9e8af52de219114c82a96b3983195436bab9f8311b50aa1d5e17524a84b8f5f7"}}