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Distribution of genus among numerical semigroups with fixed Frobenius number
A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number...
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| Published in: | Semigroup forum 2022-06, Vol.104 (3), p.704-723 |
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| creator | Singhal, Deepesh |
| description | A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number
f
and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number
f
have genus close to
3
f
4
. We denote the number of numerical semigroups of Frobenius number
f
by
N
(
f
). While
N
(
f
) is not monotonic we prove that
N
(
f
)
<
N
(
f
+
2
)
for every
f
. |
| doi_str_mv | 10.1007/s00233-022-10282-6 |
| format | article |
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f
and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number
f
have genus close to
3
f
4
. We denote the number of numerical semigroups of Frobenius number
f
by
N
(
f
). While
N
(
f
) is not monotonic we prove that
N
(
f
)
<
N
(
f
+
2
)
for every
f
.</description><identifier>ISSN: 0037-1912</identifier><identifier>EISSN: 1432-2137</identifier><identifier>DOI: 10.1007/s00233-022-10282-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Complement ; Mathematics ; Mathematics and Statistics ; Monoids ; Number theory ; Power series ; Research Article ; Semigroups</subject><ispartof>Semigroup forum, 2022-06, Vol.104 (3), p.704-723</ispartof><rights>The Author(s) 2022. corrected publication 2022</rights><rights>The Author(s) 2022. corrected publication 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-5a11f91934c882a11bd358d2b5d42a5e7c6c62f2fd8b1e586aeb79a0c1dc92043</citedby><cites>FETCH-LOGICAL-c363t-5a11f91934c882a11bd358d2b5d42a5e7c6c62f2fd8b1e586aeb79a0c1dc92043</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Singhal, Deepesh</creatorcontrib><title>Distribution of genus among numerical semigroups with fixed Frobenius number</title><title>Semigroup forum</title><addtitle>Semigroup Forum</addtitle><description>A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number
f
and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number
f
have genus close to
3
f
4
. We denote the number of numerical semigroups of Frobenius number
f
by
N
(
f
). While
N
(
f
) is not monotonic we prove that
N
(
f
)
<
N
(
f
+
2
)
for every
f
.</description><subject>Algebra</subject><subject>Complement</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Monoids</subject><subject>Number theory</subject><subject>Power series</subject><subject>Research Article</subject><subject>Semigroups</subject><issn>0037-1912</issn><issn>1432-2137</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kMFKxDAURYMoOI7-gKuA62jeS5u2SxmdURhwo-uQpmnNMG3GpEX9e6MV3Ll6XDj3PjiEXAK_Bs6Lm8g5CsE4IgOOJTJ5RBaQCWQIojgmC85FwaACPCVnMe54ylyKBdneuTgGV0-j8wP1Le3sMEWqez90dJh6G5zRexpt77rgp0Ok7258pa37sA1dB1_bwSU-kbUN5-Sk1ftoL37vkrys759XD2z7tHlc3W6ZEVKMLNcAbQWVyExZYgp1I_KywTpvMtS5LYw0Eltsm7IGm5dS27qoNDfQmAp5Jpbkat49BP822TiqnZ_CkF4qlKWAHLMMEoUzZYKPMdhWHYLrdfhUwNW3NTVbU8ma-rGmZCqJuRQTPHQ2_E3_0_oCldFwAQ</recordid><startdate>20220601</startdate><enddate>20220601</enddate><creator>Singhal, Deepesh</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20220601</creationdate><title>Distribution of genus among numerical semigroups with fixed Frobenius number</title><author>Singhal, Deepesh</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-5a11f91934c882a11bd358d2b5d42a5e7c6c62f2fd8b1e586aeb79a0c1dc92043</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Complement</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Monoids</topic><topic>Number theory</topic><topic>Power series</topic><topic>Research Article</topic><topic>Semigroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Singhal, Deepesh</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Semigroup forum</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Singhal, Deepesh</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Distribution of genus among numerical semigroups with fixed Frobenius number</atitle><jtitle>Semigroup forum</jtitle><stitle>Semigroup Forum</stitle><date>2022-06-01</date><risdate>2022</risdate><volume>104</volume><issue>3</issue><spage>704</spage><epage>723</epage><pages>704-723</pages><issn>0037-1912</issn><eissn>1432-2137</eissn><abstract>A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number
f
and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number
f
have genus close to
3
f
4
. We denote the number of numerical semigroups of Frobenius number
f
by
N
(
f
). While
N
(
f
) is not monotonic we prove that
N
(
f
)
<
N
(
f
+
2
)
for every
f
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00233-022-10282-6</doi><tpages>20</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
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| ispartof | Semigroup forum, 2022-06, Vol.104 (3), p.704-723 |
| issn | 0037-1912 1432-2137 |
| language | eng |
| recordid | cdi_proquest_journals_2683152441 |
| source | Springer Link |
| subjects | Algebra Complement Mathematics Mathematics and Statistics Monoids Number theory Power series Research Article Semigroups |
| title | Distribution of genus among numerical semigroups with fixed Frobenius number |
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