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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.01212","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-05T00:55:48Z","cross_cats_sorted":["math.GR","math.NT"],"title_canon_sha256":"6cdd454542cba450d8ffc0cf0025fbdab17fb60a6781dc4839c992b4b07816db","abstract_canon_sha256":"954a7c91cd486c39ff1af529a1d5d6d060e29a89f68dad0f776197418621d349"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:09.134459Z","signature_b64":"I4EO9xW+InFoZpSJfOAqRS3anJcEXDpkf+R727FMA7SLXWE36Z3IBB6GZfJwViLcMBFKflXisbM5kW4SQYvdCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ace43692425c32769d50a6ec3c28484dcb43e1d6c4f4ad5158b2a1afda39ae4e","last_reissued_at":"2026-05-18T00:38:09.133841Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:09.133841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\gamma = \\lfloor g/3 \\rfloor +1, \\ldots, \\lfloor 2g/3\\rfloor$. 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