{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T11:20:57Z","timestamp":1779103257886,"version":"3.51.4"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T00:00:00Z","timestamp":1779062400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T00:00:00Z","timestamp":1779062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007655","name":"Czech Technical University in Prague","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007655","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Theory Comput Syst"],"published-print":{"date-parts":[[2026,6]]},"abstract":"Abstract<\/jats:title>\n \n The factor complexity\n \n \n $$\\mathcal {C}_{{\\textbf {u}}}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of a sequence\n \n \n $$\\textbf{u}= \\varvec{u}_{0}\\varvec{u}_{1}\\varvec{u}_{2} \\cdots $$<\/jats:tex-math>\n \n \n u<\/mml:mi>\n =<\/mml:mo>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n 0<\/mml:mn>\n <\/mml:msub>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n \u22ef<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n over a finite alphabet counts the number of factors of length\n n<\/jats:italic>\n occurring in\n \n \n $$\\textbf{u}$$<\/jats:tex-math>\n \n u<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , i.e.,\n \n \n $$\\mathcal {C}_\\textbf{u}{(n)} = \\#{\\mathcal {L}}_n(\\textbf{u})$$<\/jats:tex-math>\n \n \n \n C<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n #<\/mml:mo>\n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $${\\mathcal {L}}_n\\varvec{(}\\textbf{u}\\varvec{)}= {\\{}\\varvec{u}_{{i}}\\varvec{u}_{{i+1}}\\cdots \\varvec{u}_{{i+n-1}}{: i} \\in \\mathbb {N}{\\}}$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n <\/mml:mrow>\n u<\/mml:mi>\n \n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n {<\/mml:mo>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n i<\/mml:mi>\n <\/mml:msub>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \n i<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \u22ef<\/mml:mo>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \n i<\/mml:mi>\n +<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n :<\/mml:mo>\n i<\/mml:mi>\n <\/mml:mrow>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Two factors of\n \n \n $${\\mathcal {L}}_{{n}}\\varvec{(}\\textbf{u}\\varvec{)}$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n <\/mml:mrow>\n u<\/mml:mi>\n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are said to be equivalent if they are equal or one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity\n \n \n $${r}_\\textbf{u}$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n which counts the number of non-equivalent factors of\n \n \n $${\\mathcal {L}}_{{n}}\\varvec{(}\\textbf{u}\\varvec{)}$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n <\/mml:mrow>\n u<\/mml:mi>\n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . They formulated the following conjecture: a sequence\n \n \n $$\\textbf{u}$$<\/jats:tex-math>\n \n u<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is eventually periodic if and only if\n \n \n $${r}_\\textbf{u}{(n+2)} = {r}_\\textbf{u}{(n)}$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n r<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for some\n \n \n $${n} \\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Here we prove the conjecture and characterize the sequences for which\n \n \n $${r}_\\textbf{u}{(n+2)} = {r}_\\textbf{u}{(n)+1}$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n r<\/mml:mi>\n u<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for every\n \n \n $${n} \\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and also the sequences for which the equality is satisfied for every sufficiently large\n \n \n $${n} \\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s00224-026-10278-7","type":"journal-article","created":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T10:37:02Z","timestamp":1779100622000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reflection on the Reflection Complexity"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7208-4248","authenticated-orcid":false,"given":"Lubom\u00edra","family":"Dvo\u0159\u00e1kov\u00e1","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3817-2943","authenticated-orcid":false,"given":"Edita","family":"Pelantov\u00e1","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,5,18]]},"reference":[{"key":"10278_CR1","first-page":"19","volume":"84","author":"J-P Allouche","year":"2000","unstructured":"Allouche, J.-P.: Nouveaux r\u00e9sultats de transcendance de r\u00e9els \u00e0 d\u00e9veloppement non al\u00e9atoire. 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