Categorification of Some Integer Sequences and Its Applications
Categorification of real valued sequences, and in particular of integer sequences is a novel line of investigation in the theory of representation of algebras. In this theory introduced by Ringel and Fahr, numbers of a sequence are interpreted as invariants of objects of a given category. The catego...
- Autores:
-
Fernández Espinosa, Pedro Fernando
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2020
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/79501
- Palabra clave:
- 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
Brauer configurations
Brauer configuration algebra
Categorification of integer sequences
Energy of a graph
Homological ideals
Perfect matching
Theory of representation of algebras
Configuración de Brauer
Álgebra de configuración de Brauer
Categorización algebraica de sucesiones enteras
Energía de un grafo
Ideales homologicos
Emparejamientos perfectos
Teoría de representaciones de álgebras
Álgebra
Algebra
Matemáticas
Mathematics
- Rights
- openAccess
- License
- Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Categorification of Some Integer Sequences and Its Applications |
| dc.title.translated.spa.fl_str_mv |
Categorización algebraica de algunas sucesiones de números enteros y sus aplicaciones |
| title |
Categorification of Some Integer Sequences and Its Applications |
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Categorification of Some Integer Sequences and Its Applications 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas Brauer configurations Brauer configuration algebra Categorification of integer sequences Energy of a graph Homological ideals Perfect matching Theory of representation of algebras Configuración de Brauer Álgebra de configuración de Brauer Categorización algebraica de sucesiones enteras Energía de un grafo Ideales homologicos Emparejamientos perfectos Teoría de representaciones de álgebras Álgebra Algebra Matemáticas Mathematics |
| title_short |
Categorification of Some Integer Sequences and Its Applications |
| title_full |
Categorification of Some Integer Sequences and Its Applications |
| title_fullStr |
Categorification of Some Integer Sequences and Its Applications |
| title_full_unstemmed |
Categorification of Some Integer Sequences and Its Applications |
| title_sort |
Categorification of Some Integer Sequences and Its Applications |
| dc.creator.fl_str_mv |
Fernández Espinosa, Pedro Fernando |
| dc.contributor.advisor.none.fl_str_mv |
Moreno Cañadas, Agustín |
| dc.contributor.author.none.fl_str_mv |
Fernández Espinosa, Pedro Fernando |
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TERENUFIA-UNAL |
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510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas |
| topic |
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas Brauer configurations Brauer configuration algebra Categorification of integer sequences Energy of a graph Homological ideals Perfect matching Theory of representation of algebras Configuración de Brauer Álgebra de configuración de Brauer Categorización algebraica de sucesiones enteras Energía de un grafo Ideales homologicos Emparejamientos perfectos Teoría de representaciones de álgebras Álgebra Algebra Matemáticas Mathematics |
| dc.subject.proposal.eng.fl_str_mv |
Brauer configurations Brauer configuration algebra Categorification of integer sequences Energy of a graph Homological ideals Perfect matching Theory of representation of algebras |
| dc.subject.proposal.spa.fl_str_mv |
Configuración de Brauer Álgebra de configuración de Brauer Categorización algebraica de sucesiones enteras Energía de un grafo Ideales homologicos Emparejamientos perfectos Teoría de representaciones de álgebras |
| dc.subject.unesco.none.fl_str_mv |
Álgebra Algebra Matemáticas Mathematics |
| description |
Categorification of real valued sequences, and in particular of integer sequences is a novel line of investigation in the theory of representation of algebras. In this theory introduced by Ringel and Fahr, numbers of a sequence are interpreted as invariants of objects of a given category. The categorification of the Fibonacci numbers via the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver is an example of this kind of identifications. In this thesis, we follow the ideas of Ringel and Fahr to categorify several integer sequences but instead of using the 3-Kronecker quiver, we deal with a kind of algebras introduced recently by Green and Schroll called Brauer configuration algebras. Relationships between these algebras, some matrix problems and rational knots are used to interpret numbers in some integer sequences as invariants of indecomposable modules over path algebras of the 2-Kronecker quiver and the four subspace quiver. The results enable us to define the message of a Brauer Configuration and labeled Brauer configurations in order to give an interpretation of the number of perfect matchings of snake graphs, the number of homological ideals of some Nakayama algebras, and the number of k-paths linking two fixed points (associated to the Lindström problem) in a quiver as specializations of indecomposable modules over suitable Brauer configuration algebras. Actually, this setting can be also used to define the Gutman index of a tree (or the trace norm of a digraph, which is a fundamental notion in the topological index theory), magic squares, and different parameters of traffic flow models in terms of this kind of algebras. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017. |
| publishDate |
2020 |
| dc.date.issued.none.fl_str_mv |
2020-07-30 |
| dc.date.accessioned.none.fl_str_mv |
2021-05-11T19:27:56Z |
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2021-05-11T19:27:56Z |
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Trabajo de grado - Doctorado |
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info:eu-repo/semantics/doctoralThesis |
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info:eu-repo/semantics/acceptedVersion |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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eng |
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eng |
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[1] N. Agudelo, J.A. de la Peña, and J.P. Rada, Extremal values of the trace norm over oriented trees, Linear Algebra Appl 505 (2016), 261-268. [2] M. Ahmed, Algebraic Combinatorics of Magic Squares, California University, Dissertation, 2004. 1-92. [3] G. Andrews, The Theory of Partitions, Cambridge University. Press, Cambridge, 1998. 1-255. [4] M. Armenta, Homological Ideals of Finite Dimensional Algebras, CIMAT, Mexico, 2016. Master Thesis. [5] I. Assem, D. Simson, and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Cambridge University Press, Cambridge UK, 2006. 1-457. [6] M. Auslander, M. I. Platzeck, and G. Todorov, Homological Theory of Idempotent Ideals, Transactions of the American Mathematical Society 332 (1992), no. 2, 667-692. [7] M. Auslander, I. Reiten, and O. Smalo , Representation Theory of Artin Algebras, Cambridge University Press, Cambridge UK, 1997. 1-425. [8] D. J. Benson, Representations and cohomology of finite groups I, Vol. 30, Cambridge Studies in Advanced Mathematics, Cambridge, 1991. 1-246. [9] V.M. Bondarenko, Representations of bundles of semichained sets and their applications, Algebra i Analiz 3 (1991), no. 5, 38-61. English Translation; W. Crawley-Boevey, U. Hansper, I. Voulis, 2018. [10] S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967), 100-114. [11] S. Brenner, On four subspaces of a vector space, J. Algebra 29 (1974), 587-599. [12] N.D. Cahill, J.R. D'Errico, D.A. Narayan, and J.Y. Narayan, Fibonacci determinants, The college mathematics journal 33 (2002), no. 3, 221-225. [13] A. M. Cañadas, P.F.F. Espinosa, and I.D.M. Gaviria, Categorification of some integer sequences via Kronecker modules, JPANTA 38 (2016), no. 4, 339-347. [14] A. M. Cañadas, H. Giraldo, and G.B. Rios, An algebraic approach to the number of some antichains in the powerset 2n, JPANTA 38 (2016), no. 1, 45-62. [15] E. Chen, Topics in Combinatorics; Lecture Notes, MIT, Massachusetts, 2017. 1-40. [16] J.H. Conway, An enumeration of knots and links and some of their algebraic properties, Proceedings of the conference on Computational problems in Abstract Algebra held at Oxford in 1967, Pergamon Press, J. Leech ed. (1970), 329-358. [17] J. A. De la Peña and Changchang Xi, Hochschild Cohomology of Algebras with Homological Ideals, Tsukuba J. Math. 30 (2006), no. 1, 61-79. [18] D.Z. Djokovic, Classification of pairs consisting of a linear and a semi-linear map, Linear Algebra Appl. 20 (1978), 147-165. [19] P. Fahr and C. M. Ringel, A partition formula for Fibonacci numbers, J. Integer Seq. 11 (2008), no. 08.14, 1-9. [20] P. Fahr and C. M. Ringel, Categorification of the Fibonacci numbers using representations of quivers, J. Integer Seq. 15 (2012), no. 12.2.1, 1-12. [21] P. Fahr and C. M. Ringel, The Fibonacci triangles, Advances in Mathematics. 230 (2012), 2513-2535. [22] S. Fomin and A. Zelevinsky, Cluster Algebras I. Foundations, J. Amer. Math. Soc 15 (2002), no. 2, 497{529 (electronic). MR 1887642 (2003f:16050). [23] S. Fomin and A. Zelevinsky, Cluster Algebras II. Finite type classification, Invent. Math 154 (2003), no. 1, 63{121. MR 2004457 (2004m:17011). [24] P. Gabriel and J.A. Peña, Quotients of representation-finite algebras, Communications in Algebra 15 (1987), 279-307. [25] P. Gabriel and A.V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopedia of Math. Sc., vol. 73, Springer-Verlag, 1992. 1-177. [26] M.A. Gatica, M. Lanzilotta, and M.I. Platzeck, Idempotent Ideals and the Igusa-Todorov Functions, Algebra Represent Theory 20 (2017), 275-287. [27] I.D. M. Gaviria, The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications, (PhD. Thesis) Universidad Nacional de Colombia (2020), 1-164. [28] I.M. Gelfand and V.A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite dimensional vector space, Colloq. Math. Soc. Janos Bolyai, Hilbert Space Operators, Tihany 5 (1970), 163-237. [29] I. Gessel and X.G. Viennot, Determinants, paths and plane partitions, preprint (1989). [30] E.L. Green and S. Schroll, Almost gentle algebras and their trivial extensions, Proceedings of the Edinburgh Mathematical Society (2018), 1-16. [31] E.L. Green and S. Schroll, Brauer configuration algebras: A generalization of Brauer graph algebras, Bull. Sci. Math. 141 (2017), 539-572. [32] B. Guberfain, R. Nasser, M. Casanova, and H. Lopes, BusesInRio: buses as mobile traffic sensors Managing the bus GPS data in the City of Rio de Janeiro, 17th IEEE International Conference on Mobile Data Management (2016), 369-372. [33] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978), 1-22. [34] R.K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004. 1-161. [35] D. Happel and D. Zacharia, Algebras of finite global dimension, 8, Springer, Heidelberg, 2013. In: Algebras, quivers and representations, Abel Symp. [36] L. Hille and D. Ploog, Exceptional sequences and spherical modules for the Auslander algebra of k[x]=(x^t), arXiv 1709.03618v2 (2017), 1-19. [37] L. Hille and D. Ploog, Tilting chains of negative curves on rational surfaces, Nagoya Math Journal (2017), 1-16. [38] L. Hille and G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1999), 35-52. [39] A. Hubery and H. Krause, A categorification of non-crossing partitions and representations of quivers, arXiv 1310.1907 (2013), 1-34. [40] C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers, Comp. Math 145 (2009), 1533-1562. [41] L. H. Kauffman and S. Lambropoulou, Classifying and applying rational knots and rational tangles, 2002. 1-37. [42] D. Knuth, The Art of Computer Programming, Vol. 4, Addison-Wesley, 2004. 1-300. [43] G. Kreweras, Sur les partitions non croisées d. un cycle, Discrete Math 1 (1972), no. 4, 333-350. [44] P. Lampe, Cluster Algebras, Preprint (2013), 1-64. [45] R.B. Lin, On the applications of partition diagrams for integer partitioning, Proc. The 23rd workshop on combinatorial mathematics and computation theory (2006), 349-354. [46] P. Luschny, Counting with partitions, 2011. http://www.luschny.de/math/seq/CountingWithPartitions.html. [47] M.E. Mays and J. Wojciechowski, A determinant property of Catalan numbers, Discrete Mathematics, 211 (2000), 125-133. [48] G. Musiker, R. Schiffler, and L. Williams, Positivity for cluster algebras, Adv. Math 227 (2011), 2241-2308. [49] L.A. Nazarova, Representations of a tetrad, Izv. AN SSSR Ser. Mat. 7 (1967), no. 4, 1361-1378 (in Russian). English transl. in: Math. USSR Izvestija 1 (1967) 1305-1321, 1969. [50] L.A. Nazarova, Representations of quivers of infinite type, Izv. AN SSSR Ser. Mat. 37 (1973), 752-791 (in Russian). English transl. in: Math. USSR Izvestija 7 (1973) 749-792. [51] L.A. Nazarova and A.V. Roiter, On the Problem of I.M. Gelfand, Funct. Anal. Appl. 31 (1973), no. 6, 54-69. [52] L.A. Nazarova and A.V. Roiter, Representations of partially ordered sets, Zap. Nauchn. Semin. LOMI 28 (1972), 5-31 (in Russian). English transl. in J. Sov. Math. 3 (1975) 585-606. [53] J. Prop, The combinatorics of frieze patterns and Markoff numbers, arXiv 4 (2008), no. math/0511633, 1-12. [54] C.M. Ringel, The Catalan combinatorics of the hereditary artin algebras, Contemporary Mathematics, 673 (2016), 51-177. [55] A. Ripatti, On the number of semi-magic squares of order 6, arXiv 1807.02983v1 (2017), 1-14. [56] R. Schiffler and I. Canackci, Snake graphs and continued fractions, European J. Combin. 86 (2020), 1-19. [57] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces, J. Algebra 382 (2013), 240-281. [58] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces II: Self-crossings snake graphs, Math. Z. 281 (2015), no. 1, 55-102. [59] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces III: Band graphs and snake rings, Int. Math. Res. Not. (IMRN) 157 (2017), 1-82. [60] R. Schiffler and I. Canackci, Cluster algebras and continued fractions, Compositio Mathematica 154 (2018), no. 3, 565-593. [61] S. Schroll and I. Canackci, Lattice bijections for string modules snake graphs and the weak Bruhat order, arXiv 1 (2018), no. 1811.06064. [62] H. Schubert, Knoten mit zwei Brücken, Math. Zeitschrift 65 (1956), 133-170. [63] W. Shi, Q. Kong, and Y. Liu, A GPS/GIS Integrated System for Urban Traffic Flow Analysis, Proceedings of the 11th International IEEE Conference on Intelligent Transportation Systems (2008), 844-849. [64] A. Sierra, The dimension of the center of a Brauer configuration algebra, J. Algebra 510 (2018), 289-318. [65] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Gordon and Breach, London, 1992. [66] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation, Available at https://oeis.org. [67] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A033812, The OEIS Foundation. [68] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A080992, The OEIS Foundation. [69] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A006052, The OEIS Foundation. [70] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A100705, The OEIS Foundation. [71] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A052558, The OEIS Foundation. [72] R.P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Mathematical Journal 43 (1976), no. 3, 511-531. [73] B.M. Stewart, Magic graphs, Canad, J. Math 18 (1966), 1031-1059. [74] A.G. Zavadskij and G. Medina, The four subspace problem; An elementary solution, Linear Algebra Appl. 392 (2004), 11-23. [75] A.V. Zabarilo and A.G. Zavadskij, One-Parameter Equipped Posets and Their Representations, Functional Analysis and Its Applications 34 (2000), no. 2, 138-140. [76] A.G. Zavadskij, On the Kronecker problem and related problems of linear algebra, Linear Algebra Appl. 425 (2007), 26-62. |
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Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Moreno Cañadas, Agustín9ca55eaf75ecd87559010093e719d1f8Fernández Espinosa, Pedro Fernandod23dc75e3f32b467caef072581ac5972TERENUFIA-UNAL2021-05-11T19:27:56Z2021-05-11T19:27:56Z2020-07-30https://repositorio.unal.edu.co/handle/unal/79501Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/Categorification of real valued sequences, and in particular of integer sequences is a novel line of investigation in the theory of representation of algebras. In this theory introduced by Ringel and Fahr, numbers of a sequence are interpreted as invariants of objects of a given category. The categorification of the Fibonacci numbers via the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver is an example of this kind of identifications. In this thesis, we follow the ideas of Ringel and Fahr to categorify several integer sequences but instead of using the 3-Kronecker quiver, we deal with a kind of algebras introduced recently by Green and Schroll called Brauer configuration algebras. Relationships between these algebras, some matrix problems and rational knots are used to interpret numbers in some integer sequences as invariants of indecomposable modules over path algebras of the 2-Kronecker quiver and the four subspace quiver. The results enable us to define the message of a Brauer Configuration and labeled Brauer configurations in order to give an interpretation of the number of perfect matchings of snake graphs, the number of homological ideals of some Nakayama algebras, and the number of k-paths linking two fixed points (associated to the Lindström problem) in a quiver as specializations of indecomposable modules over suitable Brauer configuration algebras. Actually, this setting can be also used to define the Gutman index of a tree (or the trace norm of a digraph, which is a fundamental notion in the topological index theory), magic squares, and different parameters of traffic flow models in terms of this kind of algebras. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.La categorización de sucesiones de números reales, y en particular de sucesiones enteras es una nueva línea de investigación en la teoría de la representación de álgebras. En esta teoría introducida por Ringel y Fahr, los números de una sucesión se interpretan como invariantes de objetos de una categoría dada. La categorización de los números de Fibonacci vía la estructura del carcaj de Auslander-Reiten del carcaj 3-Kronecker es un ejemplo de este tipo de identificaciones. En esta tesis, seguimos las ideas de Ringel y Fahr para categorizar sucesiones de números enteros pero en lugar de utilizar el carcaj 3-Kronecker nosotros usamos un tipo de álgebras introducidas recientemente por Green y Schroll llamadas álgebras de configuración de Brauer. Las relaciones entre estas álgebras, algunos problemas matriciales y nudos racionales se utilizan para interpretar números en algunas secuencias enteras como invariantes de módulos indescomponibles sobre el álgebra de caminos del carcaj 2-Kronecker y el carcaj de los cuatro subespacios. Los resultados nos permiten definir el mensaje de una configuración de Brauer y configuraciones de Brauer etiquetadas para dar una interpretación del número de emparejamientos perfectos de los gráficos de serpientes, el número de ideales homológicos de algunas álgebras de Nakayama y el número de k-trayectorias que enlazan dos puntos fijos (asociados al problema de Lindström) en un carcaj como especializaciones de módulos indescomponibles sobre álgebras de configuración de Brauer adecuadas. En realidad, este tipo de configuraciones también se pueden utilizar para definir el índice de Gutman de un árbol (o la norma traza de un dígrafo, que es una noción fundamental en la teoría del índice topológico), cuadrados mágicos y diferentes parámetros de los modelos de flujo de tráfico en términos de este tipo de álgebras. Esta investigación fue apoyada parcialmente por COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.DoctoradoTeoría de representaciones de álgebras y sus aplicaciones1 recurso en línea (142 páginas)application/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotáUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::519 - Probabilidades y matemáticas aplicadasBrauer configurationsBrauer configuration algebraCategorification of integer sequencesEnergy of a graphHomological idealsPerfect matchingTheory of representation of algebrasConfiguración de BrauerÁlgebra de configuración de BrauerCategorización algebraica de sucesiones enterasEnergía de un grafoIdeales homologicosEmparejamientos perfectosTeoría de representaciones de álgebrasÁlgebraAlgebraMatemáticasMathematicsCategorification of Some Integer Sequences and Its ApplicationsCategorización algebraica de algunas sucesiones de números enteros y sus aplicacionesTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TD[1] N. Agudelo, J.A. de la Peña, and J.P. Rada, Extremal values of the trace norm over oriented trees, Linear Algebra Appl 505 (2016), 261-268.[2] M. Ahmed, Algebraic Combinatorics of Magic Squares, California University, Dissertation, 2004. 1-92.[3] G. Andrews, The Theory of Partitions, Cambridge University. Press, Cambridge, 1998. 1-255.[4] M. Armenta, Homological Ideals of Finite Dimensional Algebras, CIMAT, Mexico, 2016. Master Thesis.[5] I. Assem, D. Simson, and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Cambridge University Press, Cambridge UK, 2006. 1-457.[6] M. Auslander, M. I. Platzeck, and G. Todorov, Homological Theory of Idempotent Ideals, Transactions of the American Mathematical Society 332 (1992), no. 2, 667-692.[7] M. Auslander, I. Reiten, and O. Smalo , Representation Theory of Artin Algebras, Cambridge University Press, Cambridge UK, 1997. 1-425.[8] D. J. Benson, Representations and cohomology of finite groups I, Vol. 30, Cambridge Studies in Advanced Mathematics, Cambridge, 1991. 1-246.[9] V.M. Bondarenko, Representations of bundles of semichained sets and their applications, Algebra i Analiz 3 (1991), no. 5, 38-61. English Translation; W. Crawley-Boevey, U. Hansper, I. Voulis, 2018.[10] S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967), 100-114.[11] S. Brenner, On four subspaces of a vector space, J. Algebra 29 (1974), 587-599.[12] N.D. Cahill, J.R. D'Errico, D.A. Narayan, and J.Y. Narayan, Fibonacci determinants, The college mathematics journal 33 (2002), no. 3, 221-225.[13] A. M. Cañadas, P.F.F. Espinosa, and I.D.M. Gaviria, Categorification of some integer sequences via Kronecker modules, JPANTA 38 (2016), no. 4, 339-347.[14] A. M. Cañadas, H. Giraldo, and G.B. Rios, An algebraic approach to the number of some antichains in the powerset 2n, JPANTA 38 (2016), no. 1, 45-62.[15] E. Chen, Topics in Combinatorics; Lecture Notes, MIT, Massachusetts, 2017. 1-40.[16] J.H. Conway, An enumeration of knots and links and some of their algebraic properties, Proceedings of the conference on Computational problems in Abstract Algebra held at Oxford in 1967, Pergamon Press, J. Leech ed. (1970), 329-358.[17] J. A. De la Peña and Changchang Xi, Hochschild Cohomology of Algebras with Homological Ideals, Tsukuba J. Math. 30 (2006), no. 1, 61-79.[18] D.Z. Djokovic, Classification of pairs consisting of a linear and a semi-linear map, Linear Algebra Appl. 20 (1978), 147-165.[19] P. Fahr and C. M. Ringel, A partition formula for Fibonacci numbers, J. Integer Seq. 11 (2008), no. 08.14, 1-9.[20] P. Fahr and C. M. Ringel, Categorification of the Fibonacci numbers using representations of quivers, J. Integer Seq. 15 (2012), no. 12.2.1, 1-12.[21] P. Fahr and C. M. Ringel, The Fibonacci triangles, Advances in Mathematics. 230 (2012), 2513-2535.[22] S. Fomin and A. Zelevinsky, Cluster Algebras I. Foundations, J. Amer. Math. Soc 15 (2002), no. 2, 497{529 (electronic). MR 1887642 (2003f:16050).[23] S. Fomin and A. Zelevinsky, Cluster Algebras II. Finite type classification, Invent. Math 154 (2003), no. 1, 63{121. MR 2004457 (2004m:17011).[24] P. Gabriel and J.A. Peña, Quotients of representation-finite algebras, Communications in Algebra 15 (1987), 279-307.[25] P. Gabriel and A.V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopedia of Math. Sc., vol. 73, Springer-Verlag, 1992. 1-177.[26] M.A. Gatica, M. Lanzilotta, and M.I. Platzeck, Idempotent Ideals and the Igusa-Todorov Functions, Algebra Represent Theory 20 (2017), 275-287.[27] I.D. M. Gaviria, The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications, (PhD. Thesis) Universidad Nacional de Colombia (2020), 1-164.[28] I.M. Gelfand and V.A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite dimensional vector space, Colloq. Math. Soc. Janos Bolyai, Hilbert Space Operators, Tihany 5 (1970), 163-237.[29] I. Gessel and X.G. Viennot, Determinants, paths and plane partitions, preprint (1989).[30] E.L. Green and S. Schroll, Almost gentle algebras and their trivial extensions, Proceedings of the Edinburgh Mathematical Society (2018), 1-16.[31] E.L. Green and S. Schroll, Brauer configuration algebras: A generalization of Brauer graph algebras, Bull. Sci. Math. 141 (2017), 539-572.[32] B. Guberfain, R. Nasser, M. Casanova, and H. Lopes, BusesInRio: buses as mobile traffic sensors Managing the bus GPS data in the City of Rio de Janeiro, 17th IEEE International Conference on Mobile Data Management (2016), 369-372.[33] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978), 1-22.[34] R.K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004. 1-161.[35] D. Happel and D. Zacharia, Algebras of finite global dimension, 8, Springer, Heidelberg, 2013. In: Algebras, quivers and representations, Abel Symp.[36] L. Hille and D. Ploog, Exceptional sequences and spherical modules for the Auslander algebra of k[x]=(x^t), arXiv 1709.03618v2 (2017), 1-19.[37] L. Hille and D. Ploog, Tilting chains of negative curves on rational surfaces, Nagoya Math Journal (2017), 1-16.[38] L. Hille and G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1999), 35-52.[39] A. Hubery and H. Krause, A categorification of non-crossing partitions and representations of quivers, arXiv 1310.1907 (2013), 1-34.[40] C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers, Comp. Math 145 (2009), 1533-1562.[41] L. H. Kauffman and S. Lambropoulou, Classifying and applying rational knots and rational tangles, 2002. 1-37.[42] D. Knuth, The Art of Computer Programming, Vol. 4, Addison-Wesley, 2004. 1-300.[43] G. Kreweras, Sur les partitions non croisées d. un cycle, Discrete Math 1 (1972), no. 4, 333-350.[44] P. Lampe, Cluster Algebras, Preprint (2013), 1-64.[45] R.B. Lin, On the applications of partition diagrams for integer partitioning, Proc. The 23rd workshop on combinatorial mathematics and computation theory (2006), 349-354.[46] P. Luschny, Counting with partitions, 2011. http://www.luschny.de/math/seq/CountingWithPartitions.html.[47] M.E. Mays and J. Wojciechowski, A determinant property of Catalan numbers, Discrete Mathematics, 211 (2000), 125-133.[48] G. Musiker, R. Schiffler, and L. Williams, Positivity for cluster algebras, Adv. Math 227 (2011), 2241-2308.[49] L.A. Nazarova, Representations of a tetrad, Izv. AN SSSR Ser. Mat. 7 (1967), no. 4, 1361-1378 (in Russian). English transl. in: Math. USSR Izvestija 1 (1967) 1305-1321, 1969.[50] L.A. Nazarova, Representations of quivers of infinite type, Izv. AN SSSR Ser. Mat. 37 (1973), 752-791 (in Russian). English transl. in: Math. USSR Izvestija 7 (1973) 749-792.[51] L.A. Nazarova and A.V. Roiter, On the Problem of I.M. Gelfand, Funct. Anal. Appl. 31 (1973), no. 6, 54-69.[52] L.A. Nazarova and A.V. Roiter, Representations of partially ordered sets, Zap. Nauchn. Semin. LOMI 28 (1972), 5-31 (in Russian). English transl. in J. Sov. Math. 3 (1975) 585-606.[53] J. Prop, The combinatorics of frieze patterns and Markoff numbers, arXiv 4 (2008), no. math/0511633, 1-12.[54] C.M. Ringel, The Catalan combinatorics of the hereditary artin algebras, Contemporary Mathematics, 673 (2016), 51-177.[55] A. Ripatti, On the number of semi-magic squares of order 6, arXiv 1807.02983v1 (2017), 1-14.[56] R. Schiffler and I. Canackci, Snake graphs and continued fractions, European J. Combin. 86 (2020), 1-19.[57] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces, J. Algebra 382 (2013), 240-281.[58] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces II: Self-crossings snake graphs, Math. Z. 281 (2015), no. 1, 55-102.[59] R. Schiffler and I. Canackci, Snake graphs calculus and cluster algebras from surfaces III: Band graphs and snake rings, Int. Math. Res. Not. (IMRN) 157 (2017), 1-82.[60] R. Schiffler and I. Canackci, Cluster algebras and continued fractions, Compositio Mathematica 154 (2018), no. 3, 565-593.[61] S. Schroll and I. Canackci, Lattice bijections for string modules snake graphs and the weak Bruhat order, arXiv 1 (2018), no. 1811.06064.[62] H. Schubert, Knoten mit zwei Brücken, Math. Zeitschrift 65 (1956), 133-170.[63] W. Shi, Q. Kong, and Y. Liu, A GPS/GIS Integrated System for Urban Traffic Flow Analysis, Proceedings of the 11th International IEEE Conference on Intelligent Transportation Systems (2008), 844-849.[64] A. Sierra, The dimension of the center of a Brauer configuration algebra, J. Algebra 510 (2018), 289-318.[65] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Gordon and Breach, London, 1992.[66] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation, Available at https://oeis.org.[67] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A033812, The OEIS Foundation.[68] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A080992, The OEIS Foundation.[69] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A006052, The OEIS Foundation.[70] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A100705, The OEIS Foundation.[71] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, Vol. http://oeis.org/A052558, The OEIS Foundation.[72] R.P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Mathematical Journal 43 (1976), no. 3, 511-531.[73] B.M. Stewart, Magic graphs, Canad, J. Math 18 (1966), 1031-1059.[74] A.G. Zavadskij and G. Medina, The four subspace problem; An elementary solution, Linear Algebra Appl. 392 (2004), 11-23.[75] A.V. Zabarilo and A.G. Zavadskij, One-Parameter Equipped Posets and Their Representations, Functional Analysis and Its Applications 34 (2000), no. 2, 138-140.[76] A.G. Zavadskij, On the Kronecker problem and related problems of linear algebra, Linear Algebra Appl. 425 (2007), 26-62.Categorification of Some Integer Sequences and Its ApplicationsCOLCIENCIASORIGINAL1022993409.2020.pdf1022993409.2020.pdfTesis de Doctorado en Ciencias Matemáticasapplication/pdf2885056https://repositorio.unal.edu.co/bitstream/unal/79501/1/1022993409.2020.pdf45fa7f584cdd7efd95e3ba591f54a29cMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/79501/2/license.txtcccfe52f796b7c63423298c2d3365fc6MD52CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://repositorio.unal.edu.co/bitstream/unal/79501/3/license_rdf4460e5956bc1d1639be9ae6146a50347MD53THUMBNAIL1022993409.2020.pdf.jpg1022993409.2020.pdf.jpgGenerated Thumbnailimage/jpeg3941https://repositorio.unal.edu.co/bitstream/unal/79501/4/1022993409.2020.pdf.jpg23aad2f61742e87bbc2fd5520278ea8fMD54unal/79501oai:repositorio.unal.edu.co:unal/795012023-07-29 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