{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:GCCANJLTDLIVOYWBI7MUB2AIJ6","short_pith_number":"pith:GCCANJLT","canonical_record":{"source":{"id":"1801.00416","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-01T09:20:19Z","cross_cats_sorted":[],"title_canon_sha256":"71d76182ffa82c6bc648e7cd7cfaf15a97d277f6b5cb05377f7b271525126f4f","abstract_canon_sha256":"ab621fa507d143c3614906a7ffaa7195d30a7aea0f11ce745fac4bbb09e305bb"},"schema_version":"1.0"},"canonical_sha256":"308406a5731ad15762c147d940e8084f94b4760262624ed7a854f687c4df12d5","source":{"kind":"arxiv","id":"1801.00416","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00416","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00416v4","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00416","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"pith_short_12","alias_value":"GCCANJLTDLIV","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GCCANJLTDLIVOYWB","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GCCANJLT","created_at":"2026-05-18T12:32:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:GCCANJLTDLIVOYWBI7MUB2AIJ6","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00416","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-01T09:20:19Z","cross_cats_sorted":[],"title_canon_sha256":"71d76182ffa82c6bc648e7cd7cfaf15a97d277f6b5cb05377f7b271525126f4f","abstract_canon_sha256":"ab621fa507d143c3614906a7ffaa7195d30a7aea0f11ce745fac4bbb09e305bb"},"schema_version":"1.0"},"canonical_sha256":"308406a5731ad15762c147d940e8084f94b4760262624ed7a854f687c4df12d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:04.817326Z","signature_b64":"vp0xL7JQM4vUoxvAukwG+cMJGi3DloXzRwDAclsoDa93J5Q8u6WVre9kkPJEn4qNmAIwt20x3T6g1FgNsM8HDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"308406a5731ad15762c147d940e8084f94b4760262624ed7a854f687c4df12d5","last_reissued_at":"2026-05-17T23:45:04.816678Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:04.816678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00416","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:45:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SCq85ZAmd0P1os2DbC9HvsnqLIYwUoiP4Rh2UsxfdcIFp9AkB6n6LlJpYaNy8S3xFoDRJToAVk4Akp4bbgvQAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T13:10:31.705694Z"},"content_sha256":"70724398f06cd616bbc8c168a90ab70bb44001dddf52951f9edc83e97aaf00c7","schema_version":"1.0","event_id":"sha256:70724398f06cd616bbc8c168a90ab70bb44001dddf52951f9edc83e97aaf00c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:GCCANJLTDLIVOYWBI7MUB2AIJ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Permutations with small maximal $k$-consecutive sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani, Kazuki Kurimoto","submitted_at":"2018-01-01T09:20:19Z","abstract_excerpt":"Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(\\pi_1,\\ldots,\\pi_n)$ of integers $1,\\ldots,n$, we consider $k$-consecutive sums of $\\pi$, i.e., $s_i:=\\sum_{j=0}^{k-1}\\pi_{i+j}$ for $i=1,\\ldots,n$, where we let $\\pi_{n+j}=\\pi_j$. What we want to do in this paper is to know the exact value of $$\\mathrm{msum}(n,k):=\\min\\left\\{\\max\\{s_i : i=1,\\ldots,n\\} -\\frac{k(n+1)}{2}: \\pi \\in S_n\\right\\},$$ where $S_n$ denotes the set of all permutations of $1,\\ldots,n$. In this paper, we determine the exact values of $\\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00416","kind":"arxiv","version":4},"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":"2026-05-17T23:45:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"esIq8sZSg6B9pebQofG2tN2jFdHySCFJS1g8vAPBFlfFb1hVrnAFPmc+M662/Fq/Wdpfm7rYlQnhz4e73n+0Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T13:10:31.706397Z"},"content_sha256":"7a9380515f67a4166b635b79a812b659f6ff6dc12642d2d0929b9b6f1e00d983","schema_version":"1.0","event_id":"sha256:7a9380515f67a4166b635b79a812b659f6ff6dc12642d2d0929b9b6f1e00d983"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/bundle.json","state_url":"https://pith.science/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/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":"2026-05-23T13:10:31Z","links":{"resolver":"https://pith.science/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6","bundle":"https://pith.science/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/bundle.json","state":"https://pith.science/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GCCANJLTDLIVOYWBI7MUB2AIJ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GCCANJLTDLIVOYWBI7MUB2AIJ6","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":"ab621fa507d143c3614906a7ffaa7195d30a7aea0f11ce745fac4bbb09e305bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-01T09:20:19Z","title_canon_sha256":"71d76182ffa82c6bc648e7cd7cfaf15a97d277f6b5cb05377f7b271525126f4f"},"schema_version":"1.0","source":{"id":"1801.00416","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00416","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00416v4","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00416","created_at":"2026-05-17T23:45:04Z"},{"alias_kind":"pith_short_12","alias_value":"GCCANJLTDLIV","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GCCANJLTDLIVOYWB","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GCCANJLT","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:7a9380515f67a4166b635b79a812b659f6ff6dc12642d2d0929b9b6f1e00d983","target":"graph","created_at":"2026-05-17T23:45:04Z","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 $n$ and $k$ be positive integers with $n>k$. Given a permutation $(\\pi_1,\\ldots,\\pi_n)$ of integers $1,\\ldots,n$, we consider $k$-consecutive sums of $\\pi$, i.e., $s_i:=\\sum_{j=0}^{k-1}\\pi_{i+j}$ for $i=1,\\ldots,n$, where we let $\\pi_{n+j}=\\pi_j$. What we want to do in this paper is to know the exact value of $$\\mathrm{msum}(n,k):=\\min\\left\\{\\max\\{s_i : i=1,\\ldots,n\\} -\\frac{k(n+1)}{2}: \\pi \\in S_n\\right\\},$$ where $S_n$ denotes the set of all permutations of $1,\\ldots,n$. In this paper, we determine the exact values of $\\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a co","authors_text":"Akihiro Higashitani, Kazuki Kurimoto","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-01T09:20:19Z","title":"Permutations with small maximal $k$-consecutive sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00416","kind":"arxiv","version":4},"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":"sha256:70724398f06cd616bbc8c168a90ab70bb44001dddf52951f9edc83e97aaf00c7","target":"record","created_at":"2026-05-17T23:45:04Z","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":"ab621fa507d143c3614906a7ffaa7195d30a7aea0f11ce745fac4bbb09e305bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-01T09:20:19Z","title_canon_sha256":"71d76182ffa82c6bc648e7cd7cfaf15a97d277f6b5cb05377f7b271525126f4f"},"schema_version":"1.0","source":{"id":"1801.00416","kind":"arxiv","version":4}},"canonical_sha256":"308406a5731ad15762c147d940e8084f94b4760262624ed7a854f687c4df12d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"308406a5731ad15762c147d940e8084f94b4760262624ed7a854f687c4df12d5","first_computed_at":"2026-05-17T23:45:04.816678Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:04.816678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vp0xL7JQM4vUoxvAukwG+cMJGi3DloXzRwDAclsoDa93J5Q8u6WVre9kkPJEn4qNmAIwt20x3T6g1FgNsM8HDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:04.817326Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00416","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70724398f06cd616bbc8c168a90ab70bb44001dddf52951f9edc83e97aaf00c7","sha256:7a9380515f67a4166b635b79a812b659f6ff6dc12642d2d0929b9b6f1e00d983"],"state_sha256":"1c75eb02a440c8e14104ecf8dff97661b42ac3f10935be002c7ef43c2065d133"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V3h9zUngoMxLSobM8DdTkCGxSA5HOtGjbtPFW25jJDAPK5u1z+6fgH45zHVXuFv1wv2IYjExGWC7KLRD37q6BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T13:10:31.710508Z","bundle_sha256":"f70bd66c36012ed1b925c20c3b0d59e0bd1607b50b2dd2dcc7c6377a56e4d29c"}}