{"paper":{"title":"Counting numerical semigroups by genus and even gaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Fernando Torres, Matheus Bernardini","submitted_at":"2016-12-05T00:55:48Z","abstract_excerpt":"Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_\\gamma(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\\gamma$. We show that $N_\\gamma(g)=N_\\gamma(3\\gamma)$ for $\\gamma \\leq \\lfloor g/3\\rfloor$ and $N_\\gamma(g)=0$ for $\\gamma > \\lfloor 2g/3\\rfloor$; thus the question above is true provided that $N_\\gamma(g+1) > N_\\gamma(g)$ for $\\gamma = \\lfloor g/3 \\rfloor +1, \\ldots, \\lfloor 2g/3\\rfloor$. We also show that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01212","kind":"arxiv","version":2},"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"}