{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T01:06:54Z","timestamp":1648948014839},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:p> This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators. <\/jats:p>","DOI":"10.1142\/s0218196720500496","type":"journal-article","created":{"date-parts":[[2020,6,18]],"date-time":"2020-06-18T16:03:45Z","timestamp":1592496225000},"page":"1465-1483","source":"Crossref","is-referenced-by-count":0,"title":["On the enumeration and asymptotic growth of free quasigroup words"],"prefix":"10.1142","volume":"30","author":[{"given":"Jonathan D. H.","family":"Smith","sequence":"first","affiliation":[{"name":"Department of Mathematics, Iowa State University, Ames, IA 50011, USA"}]},{"given":"Stefanie G.","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, Smith College, Northampton, MA 01063, USA"}]}],"member":"219","published-online":{"date-parts":[[2020,8,8]]},"reference":[{"key":"S0218196720500496BIB001","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100027092"},{"key":"S0218196720500496BIB002","first-page":"1","volume-title":"Quasigroups and Loops: Theory and Applications","author":"Evans T.","year":"1990"},{"key":"S0218196720500496BIB003","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655"},{"key":"S0218196720500496BIB004","first-page":"939","volume":"48","author":"Grigorchuk R. I.","year":"1984","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat."},{"key":"S0218196720500496BIB005","first-page":"675","volume":"67","author":"Hall J. I.","year":"2001","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"S0218196720500496BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01109903"},{"key":"S0218196720500496BIB007","series-title":"Seminumerical Algorithms","volume-title":"The Art of Computer Programming","volume":"2","author":"Knuth D. E.","year":"1998","edition":"3"},{"key":"S0218196720500496BIB008","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004110000393"},{"key":"S0218196720500496BIB009","first-page":"151","volume":"22","author":"Sade A.","year":"1957","journal-title":"Rev. Fac. Sci. Univ. Istanbul Ser. A"},{"key":"S0218196720500496BIB010","volume-title":"An Introduction to Quasigroups and Their Representations","author":"Smith J. D. H.","year":"2007"},{"key":"S0218196720500496BIB011","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196707004323"},{"key":"S0218196720500496BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2011.08.014"},{"key":"S0218196720500496BIB013","doi-asserted-by":"publisher","DOI":"10.1201\/b15884"},{"key":"S0218196720500496BIB014","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032589"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196720500496","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,8]],"date-time":"2020-10-08T10:30:09Z","timestamp":1602153009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196720500496"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,8]]},"references-count":14,"journal-issue":{"issue":"07","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["10.1142\/S0218196720500496"],"URL":"https:\/\/doi.org\/10.1142\/s0218196720500496","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,8]]}}}