On the number of L-shapes in embedding dimension four numerical semigroups
Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes...
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| Vydané v: | Discrete mathematics Ročník 338; číslo 12; s. 2168 - 2178 |
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| Hlavní autori: | , , |
| Médium: | Journal Article Publikácia |
| Jazyk: | English |
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Elsevier B.V
06.12.2015
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| ISSN: | 0012-365X, 1872-681X |
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| Abstract | Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs.
This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups Sn, for odd n≥5, such that the number of related L-shapes is n+32. This family has her analogue to weighted Cayley digraphs of degree three.
Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. |
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| AbstractList | Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs. This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups , for odd , such that the number of related L-shapes is . This family has her analogue to weighted Cayley digraphs of degree three. Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs. This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups Sn, for odd n≥5, such that the number of related L-shapes is n+32. This family has her analogue to weighted Cayley digraphs of degree three. Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs.; This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups S-n, for odd n >= 5, such that the number of related L-shapes is n+3/2. This family has her analogue to weighted Cayley digraphs of degree three.; Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. (C) 2015 Elsevier B.V. All rights reserved. Peer Reviewed |
| Author | Aguiló-Gost, F. García-Sánchez, P.A. Llena, D. |
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| Cites_doi | 10.1016/0012-365X(94)00239-F 10.1109/TC.1987.1676963 10.37236/410 10.1016/j.endm.2014.08.025 10.1016/j.disc.2008.02.047 |
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| References | Rödseth (br000035) 1996; 148 Rosales, García-Sánchez (br000040) 2009; vol. 20 Sabariego, Santos (br000045) 2009; 309 Aguiló-Gost, García-Sánchez (br000005) 2010; 17 M. Delgado, P.A. García-Sánchez, J. Morais, “NumericalSgps”, A GAP package for numerical semigroups. Available via Fiol, Yebra, Alegre, Valero (br000025) 1987; C-36 The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.7.5; 2014 Bresinsky (br000015) 1975; 47 Aguiló-Gost, García-Sánchez, Llena (br000010) 2014; 46 . Bresinsky (10.1016/j.disc.2015.05.019_br000015) 1975; 47 Sabariego (10.1016/j.disc.2015.05.019_br000045) 2009; 309 10.1016/j.disc.2015.05.019_br000020 Aguiló-Gost (10.1016/j.disc.2015.05.019_br000010) 2014; 46 10.1016/j.disc.2015.05.019_br000030 Rosales (10.1016/j.disc.2015.05.019_br000040) 2009; vol. 20 Fiol (10.1016/j.disc.2015.05.019_br000025) 1987; C-36 Aguiló-Gost (10.1016/j.disc.2015.05.019_br000005) 2010; 17 Rödseth (10.1016/j.disc.2015.05.019_br000035) 1996; 148 |
| References_xml | – reference: M. Delgado, P.A. García-Sánchez, J. Morais, “NumericalSgps”, A GAP package for numerical semigroups. Available via – volume: 309 start-page: 1672 year: 2009 end-page: 1684 ident: br000045 article-title: Triple-loop networks with arbitrarily many minimum distance diagrams publication-title: Discrete Math. – volume: 46 start-page: 185 year: 2014 end-page: 192 ident: br000010 article-title: An algorithm to compute the primitive elements of an embedding dimension three numerical semigroups publication-title: Electron. Notes Discrete Math. – reference: The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.7.5; 2014, – volume: 148 start-page: 161 year: 1996 end-page: 173 ident: br000035 article-title: Weighted multi-connected loop networks publication-title: Discrete Math. – volume: C-36 start-page: 702 year: 1987 end-page: 713 ident: br000025 article-title: A discrete optimization problem in local networks and data alignment publication-title: IEEE Trans. Comput. – volume: 17 year: 2010 ident: br000005 article-title: Factoring in embedding dimension three numerical semigroups publication-title: Electron. J. Combin. – volume: 47 start-page: 329 year: 1975 end-page: 332 ident: br000015 article-title: On prime ideals with generic zero publication-title: Proc. Amer. Math. Soc. – volume: vol. 20 year: 2009 ident: br000040 article-title: Numerical semigroups publication-title: Developments in Mathematics – reference: . – volume: 148 start-page: 161 year: 1996 ident: 10.1016/j.disc.2015.05.019_br000035 article-title: Weighted multi-connected loop networks publication-title: Discrete Math. doi: 10.1016/0012-365X(94)00239-F – volume: C-36 start-page: 702 year: 1987 ident: 10.1016/j.disc.2015.05.019_br000025 article-title: A discrete optimization problem in local networks and data alignment publication-title: IEEE Trans. Comput. doi: 10.1109/TC.1987.1676963 – volume: vol. 20 year: 2009 ident: 10.1016/j.disc.2015.05.019_br000040 article-title: Numerical semigroups – volume: 17 issue: 1 year: 2010 ident: 10.1016/j.disc.2015.05.019_br000005 article-title: Factoring in embedding dimension three numerical semigroups publication-title: Electron. J. Combin. doi: 10.37236/410 – volume: 47 start-page: 329 year: 1975 ident: 10.1016/j.disc.2015.05.019_br000015 article-title: On prime ideals with generic zero xi=tni publication-title: Proc. Amer. Math. Soc. – volume: 46 start-page: 185 year: 2014 ident: 10.1016/j.disc.2015.05.019_br000010 article-title: An algorithm to compute the primitive elements of an embedding dimension three numerical semigroups publication-title: Electron. Notes Discrete Math. doi: 10.1016/j.endm.2014.08.025 – ident: 10.1016/j.disc.2015.05.019_br000020 – volume: 309 start-page: 1672 issue: 6 year: 2009 ident: 10.1016/j.disc.2015.05.019_br000045 article-title: Triple-loop networks with arbitrarily many minimum distance diagrams publication-title: Discrete Math. doi: 10.1016/j.disc.2008.02.047 – ident: 10.1016/j.disc.2015.05.019_br000030 |
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| SubjectTerms | 05 Combinatorics 05C Graph theory 11 Number theory 11D Diophantine equations Analogue Catenaries Classificació AMS Combinatorial analysis Combinatòria Factorization Grafs, Teoria de Graph theory Group theory L-shape LOOP NETWORKS Matemàtica discreta Matemàtiques i estadística Mathematical models Numerical semigroup Teoria de grafs Weighted Cayley digraph Àrees temàtiques de la UPC |
| Title | On the number of L-shapes in embedding dimension four numerical semigroups |
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