{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:28:55Z","timestamp":1760711335961,"version":"3.37.3"},"reference-count":21,"publisher":"EDP Sciences","license":[{"start":{"date-parts":[[2024,3,22]],"date-time":"2024-03-22T00:00:00Z","timestamp":1711065600000},"content-version":"vor","delay-in-days":81,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"INdAM-GNCS","award":["CUP E55F2200027001"],"award-info":[{"award-number":["CUP E55F2200027001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Theor. Inf. Appl."],"accepted":{"date-parts":[[2024,2,23]]},"published-print":{"date-parts":[[2024]]},"abstract":"<jats:p>We use Dyck paths having some restrictions in order to give a combinatorial interpretation for some famous number sequences. Starting from the Fibonacci numbers we show how the<jats:italic>k<\/jats:italic>-generalized Fibonacci numbers, the powers of 2, the Pell numbers, the<jats:italic>k<\/jats:italic>-generalized Pell numbers and the even-indexed Fibonacci numbers can be obtained by means of constraints on the number of consecutive valleys (at a given height) of the Dyck paths. By acting on the maximum height of the paths we get a succession of number sequences whose limit is the sequence of Catalan numbers. For these numbers we obtain a family of interesting relations including a<jats:italic>full history<\/jats:italic>recurrence relation. The whole study can be accomplished also by involving particular sets of strings<jats:italic>via<\/jats:italic>a simple encoding of Dyck paths.<\/jats:p>","DOI":"10.1051\/ita\/2024007","type":"journal-article","created":{"date-parts":[[2024,3,22]],"date-time":"2024-03-22T09:02:24Z","timestamp":1711098144000},"page":"8","source":"Crossref","is-referenced-by-count":2,"title":["Sequences from Fibonacci to Catalan: A combinatorial interpretation<i>via<\/i>Dyck paths"],"prefix":"10.1051","volume":"58","author":[{"given":"Elena","family":"Barcucci","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5363-7756","authenticated-orcid":false,"given":"Antonio","family":"Bernini","sequence":"additional","affiliation":[]},{"given":"Renzo","family":"Pinzani","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2024,3,22]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1080\/00150517.1963.12431573","volume":"1","author":"Feinberg","year":"1963","journal-title":"Fibonacci Quart."},{"key":"R2","unstructured":"Knuth D.E., The Art of Computer Programming: Sorting and Searching, 2nd edn., Vol. 3. 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Proceedings of the Third International Conference on Combinatorics, Computability and Logic, edited by Calude C.S., Dinneen M.J. and Sburlan S. (2001) 227\u2013240.","DOI":"10.1007\/978-1-4471-0717-0_19"},{"key":"R17","unstructured":"Sloane N.J.A. and the OEIS Foundation, The on-line encyclopedia of integer sequences. http:\/\/oeis.org."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Bernini A., Restricted binary strings and generalized Fibonacci numbers, in Cellular Automata and Discrete Complex Systems. AUTOMATA 2017. Lecture Notes in Computer Science, Vol. 10248, edited by Dennunzio A., Formenti E., Manzoni L. and Porreca A. (2017) 32\u201343.","DOI":"10.1007\/978-3-319-58631-1_3"},{"key":"R19","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1080\/09720529.2014.968360","volume":"18","author":"Bernini","year":"2015","journal-title":"J. Discrete Math. Sci. 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