{"paper":{"title":"Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Christopher O'Neill, Eli B. Dugan, Emi Lycan, Sarah Mendoza De La Cruz, Shadi Gaskari, Spencer Chapman, Vadim Ponomarenko","submitted_at":"2024-11-26T00:39:28Z","abstract_excerpt":"A factorization of an element $x$ in a monoid $(M, \\cdot)$ is an expression of the form $x = u_1^{z_1} \\cdots u_k^{z_k}$ for irreducible elements $u_1, \\ldots, u_k \\in M$, and the length of such a factorization is $z_1 + \\cdots + z_k$. We introduce the notion of $p$-length, a generalized notion of factorization length obtained from the $\\ell_p$-norm of the sequence $(z_1, \\ldots, z_k)$, and present asymptotic results on extremal $p$-lengths of factorizations for large elements of numerical semigroups (additive submonoids of $\\mathbb Z_{\\ge 0}$) and arithmetical congruence monoids (certain mult"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17010","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/2411.17010/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"}