{"paper":{"title":"Greedy Records and Bernstein Transfers for Fence and Circular-Fence Order Polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pyuyi Chufeng Huang","submitted_at":"2026-07-24T03:35:32Z","abstract_excerpt":"Let \\(P_\\eps\\) be the fence poset determined by an orientation \\(\\eps\\in\\{+,-\\}^{n-1}\\) of a path. We define a greedy right-to-left record statistic \\(\\rec_\\eps\\) on \\(S_n\\) and give a Bernstein-basis proof of \\[ \\sum_{\\pi\\in S_n}t^{\\rec_\\eps(\\pi)}=n!\\Omega(P_\\eps;t), \\] which for alternating signs gives the permutation statistic sought by Ferroni, Morales, and Panova for the zig-zag order polynomial. This generating function was independently obtained by Kahane; after reflection, \\(\\rec_\\eps\\) agrees pointwise with his greedy block statistic. Our transfer intertwines a continuous threshold pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22767","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.22767/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"}