{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:32:17Z","timestamp":1740133937479,"version":"3.37.3"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1800443"],"award-info":[{"award-number":["DMS-1800443"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1764366"],"award-info":[{"award-number":["DMS-1764366"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1764406"],"award-info":[{"award-number":["DMS-1764406"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["594594"],"award-info":[{"award-number":["594594"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Found. Comput. Sci."],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:p> A double occurrence word (DOW) is a word in which every symbol appears exactly twice. We define the symbol separation of a DOW [Formula: see text] to be the number of letters between the two copies of a symbol, and the separation of [Formula: see text] to be the sum of separations over all symbols in [Formula: see text]. We then analyze relationship among size, reducibility and separation of DOWs. Specifically, we provide tight bounds of separations of DOWs with a given size and characterize the words that attain those bounds. We show that all separation numbers within the bounds can be realized. We present recursive formulas for counting the numbers of DOWs with a given separation under various restrictions, such as the number of irreducible factors. These formulas can be obtained by inductive construction of all DOWs with the given separation. <\/jats:p>","DOI":"10.1142\/s0129054120500343","type":"journal-article","created":{"date-parts":[[2020,11,9]],"date-time":"2020-11-09T10:36:48Z","timestamp":1604918208000},"page":"915-928","source":"Crossref","is-referenced-by-count":0,"title":["Symbol Separation in Double Occurrence Words"],"prefix":"10.1142","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0765-9425","authenticated-orcid":false,"given":"Nata\u0161a","family":"Jonoska","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., Tampa, FL 33620, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0795-8548","authenticated-orcid":false,"given":"Masahico","family":"Saito","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., Tampa, FL 33620, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9915-9756","authenticated-orcid":false,"given":"Hwee","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon 22012, Republic of Korea"}]},{"given":"Brad","family":"Mostowski","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Florida State University, 600 W. College Ave., Tallahassee, FL 32306, USA"}]}],"member":"219","published-online":{"date-parts":[[2020,11,10]]},"reference":[{"issue":"14","key":"S0129054120500343BIB001","doi-asserted-by":"crossref","first-page":"3020","DOI":"10.1016\/j.dam.2009.06.011","volume":"157","author":"Angeleska A.","year":"2009","journal-title":"Discr. Appl. 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