{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:50:04Z","timestamp":1760237404429,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,10]],"date-time":"2020-04-10T00:00:00Z","timestamp":1586476800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"\u0420\u043e\u0441\u0441\u0438\u0439\u0441\u043a\u0438\u0439 \u0424\u043e\u043d\u0434 \u0424\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u0445 \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0439 (\u0420\u0424\u0424\u0418)","award":["18-31-00201"],"award-info":[{"award-number":["18-31-00201"]}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["18-71-00059"],"award-info":[{"award-number":["18-71-00059"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan\u2019s and Euler\u2019s triangles. On the basis of properties of Catalan\u2019s and Euler\u2019s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas.<\/jats:p>","DOI":"10.3390\/sym12040600","type":"journal-article","created":{"date-parts":[[2020,4,10]],"date-time":"2020-04-10T05:31:16Z","timestamp":1586496676000},"page":"600","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Euler\u2013Catalan\u2019s Number Triangle and Its Application"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9695-7493","authenticated-orcid":false,"given":"Yuriy","family":"Shablya","sequence":"first","affiliation":[{"name":"Department of Complex Information Security of Computer Systems, Tomsk State University of Control Systems and Radioelectronics, Tomsk 634050, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3412-432X","authenticated-orcid":false,"given":"Dmitry","family":"Kruchinin","sequence":"additional","affiliation":[{"name":"Department of Complex Information Security of Computer Systems, Tomsk State University of Control Systems and Radioelectronics, Tomsk 634050, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,10]]},"reference":[{"unstructured":"Knuth, D.E. 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