{"paper":{"title":"A Lower Bound for the Diameter of Cayley Graph of the Symmetric Group $S_n$ Generated by $(12), (12 \\dots n), (1n \\dots 2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The diameter of the Cayley graph of S_n generated by (12), (12…n), (1n…2) is at least n(n-1)/2.","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Grigorii Antiufeev","submitted_at":"2026-01-13T16:41:58Z","abstract_excerpt":"Let us denote elements of the symmetric group $S_n$ using square brackets for the one-line notation. Cycles will be represented using parentheses, following the standard cycle notation. Under this convention, the full reversal of the identity element $()$ is the element $s = [n\\ n-1 \\dots 1]$. In the present work, we obtain a lower bound on the decomposition complexity of elements $s(1n \\dots 2)^{i}$ into the generators $(12), (12 \\dots n), (1n \\dots 2)$, where $i$ ranges over the set $\\{1,2,\\dots,n\\}$. As a consequence, we derive the lower bound $n(n-1)/2$ for the diameter of Cayley graph of "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we derive the lower bound n(n-1)/2 for the diameter of Cayley graph of the group S_n generated by (12), (12 … n), (1n … 2)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a lower bound on the word length of the specific family s(1n…2)^i for i=1 to n is sufficient to lower-bound the diameter of the entire Cayley graph (i.e., that the diameter is realized or exceeded by one of these elements).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A lower bound of n(n-1)/2 is established for the diameter of the Cayley graph of S_n generated by (12), (12...n), and (1n...2).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The diameter of the Cayley graph of S_n generated by (12), (12…n), (1n…2) is at least n(n-1)/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"254dac81c3f4f7b5f948b204c5d22ff1f468700c5b721909aad0fa982e28416d"},"source":{"id":"2601.08715","kind":"arxiv","version":3},"verdict":{"id":"c9c6476f-5884-4d31-b0fc-e9f11f27324f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:46:50.435951Z","strongest_claim":"we derive the lower bound n(n-1)/2 for the diameter of Cayley graph of the group S_n generated by (12), (12 … n), (1n … 2)","one_line_summary":"A lower bound of n(n-1)/2 is established for the diameter of the Cayley graph of S_n generated by (12), (12...n), and (1n...2).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a lower bound on the word length of the specific family s(1n…2)^i for i=1 to n is sufficient to lower-bound the diameter of the entire Cayley graph (i.e., that the diameter is realized or exceeded by one of these elements).","pith_extraction_headline":"The diameter of the Cayley graph of S_n generated by (12), (12…n), (1n…2) is at least n(n-1)/2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.08715/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"}