{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:42:01Z","timestamp":1762058521133,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T00:00:00Z","timestamp":1658188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Minciencias","award":["Conv. 891"],"award-info":[{"award-number":["Conv. 891"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) T1,T2,\u2026,Tn. In this case, for 1\u2264j\u2264n\u22121, two consecutive tiles Tj and Tj+1 share exactly one edge, either the edge at the east (west) of Tj (Tj+1) or the edge at the north (south) of Tj (Tj+1). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras.<\/jats:p>","DOI":"10.3390\/computation10070124","type":"journal-article","created":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T05:37:53Z","timestamp":1658209073000},"page":"124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Snake Graphs Arising from Groves with an Application in Coding Theory"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6812-5131","authenticated-orcid":false,"given":"Agust\u00edn","family":"Moreno Ca\u00f1adas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 111321, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1386-6658","authenticated-orcid":false,"given":"Gabriel Bravo","family":"Rios","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 111321, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5858-5011","authenticated-orcid":false,"given":"Robinson-Julian","family":"Serna","sequence":"additional","affiliation":[{"name":"Escuela de Matem\u00e1ticas y Estad\u00edstica, Universidad Pedag\u00f3gica y Tecnol\u00f3gica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, Colombia"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,19]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"The combinatorics of frieze patterns and Markoff numbers","volume":"20","author":"Propp","year":"2020","journal-title":"Integers"},{"key":"ref_2","first-page":"565","article-title":"Cluster algebras and continued fractions","volume":"54","author":"Schiffler","year":"2018","journal-title":"Compos. Math."},{"key":"ref_3","first-page":"1","article-title":"Snake graphs and continued fractions","volume":"86","author":"Schiffler","year":"2020","journal-title":"Eur. J. Combin."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.jalgebra.2013.02.018","article-title":"Snake graphs calculus and cluster algebras from surfaces","volume":"382","author":"Schiffler","year":"2013","journal-title":"J. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s00209-015-1475-y","article-title":"Snake graphs calculus and cluster algebras from surfaces II: Self-crossings snake graphs","volume":"281","author":"Schiffler","year":"2015","journal-title":"Math. Z."},{"key":"ref_6","first-page":"1","article-title":"Snake graphs calculus and cluster algebras from surfaces III: Band graphs and snake rings","volume":"rnx157","author":"Schiffler","year":"2017","journal-title":"Int. Math. Res. Not. IMRN"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2241","DOI":"10.1016\/j.aim.2011.04.018","article-title":"Posiivity for cluster algebras from surfaces","volume":"227","author":"Musiker","year":"2011","journal-title":"Adv. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"102094","DOI":"10.1016\/j.aam.2020.102094","article-title":"Lattice bijections for string modules snake graphs and the weak Bruhat order","volume":"126","author":"Schroll","year":"2021","journal-title":"Adv. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.bulsci.2017.06.001","article-title":"Brauer configuration algebras: A generalization of Brauer graph algebras","volume":"121","author":"Green","year":"2017","journal-title":"Bull. Sci. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Assem, I., and Trepode, S. (2018). Brauer Graph Algebras. Homological Methods, Representation Theory, and Cluster Algebras, CRM Short Courses, Springer.","DOI":"10.1007\/978-3-319-74585-5"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ca\u00f1adas, A.M., Gaviria, I.D.M., and Vega, J.D.C. (2021). Relationships between the Chicken McNugget Problem, Mutations of Brauer Configuration Algebras and the Advanced Encryption Standard. Mathematics, 9.","DOI":"10.3390\/math9161937"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3087","DOI":"10.3934\/era.2022157","article-title":"Brauer configuration algebras defined by snake graphs and Kronecker modules","volume":"30","author":"Espinosa","year":"2022","journal-title":"Electron. Res. Arch."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Assem, I., Skowronski, A., and Simson, D. (2006). Elements of the Representation Theory of Associative Algebras, Cambridge University Press.","DOI":"10.1017\/CBO9780511614309"},{"key":"ref_14","unstructured":"Andrews, G.E. (2010). The Theory of Partitions, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.jalgebra.2018.06.002","article-title":"The dimension of the center of a Brauer configuration algebra","volume":"510","author":"Sierra","year":"2018","journal-title":"J. Algebra"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0021-8693(02)00510-0","article-title":"Arithmetree","volume":"258","author":"Loday","year":"2002","journal-title":"J. Algebra"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s10623-016-0286-4","article-title":"Energy bounds for codes and designs in Hamming spaces","volume":"82","author":"Boyvalenkov","year":"2017","journal-title":"Des. Codes Cryptogr."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/7\/124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:53:34Z","timestamp":1760140414000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/7\/124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,19]]},"references-count":17,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["computation10070124"],"URL":"https:\/\/doi.org\/10.3390\/computation10070124","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2022,7,19]]}}}