{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/near-misses-in-wilfs-conjecture","title":"Near-misses in Wilf's conjecture","arxiv_id":"1710.03623","date":"2017-10-10","proceeding":null,"authors":["Shalom Eliahou","Jean Fromentin"],"abstract":"Let S $\\subseteq$ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S $\\cap$ [0, c -- 1] and W(S) = |P||L| -- c. In 1978, Herbert Wilf asked whether W(S) $\\ge$ 0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S) $\\le$ W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S) < 0. Near-misses are very rare. Here we construct infinite families of them, with c = 4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture.","url_abs":"http://arxiv.org/abs/1710.03623v1","url_pdf":"http://arxiv.org/pdf/1710.03623v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"near-misses-in-wilfs-conjecture","repo_url":"https://github.com/hivert/NumericMonoid","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}