{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T08:54:07Z","timestamp":1767603247652,"version":"3.48.0"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T00:00:00Z","timestamp":1766620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper studies higher-order Eulerian numbers based on Stirling permutations and utilizing Eulerian triangles. It primarily focuses on the chain of higher-order Eulerian numbers, higher-order Eulerian polynomials, and higher-order Eulerian fractions, especially their computation. Many results for Eulerian numbers and second-order Eulerian numbers are generalized to higher-order Eulerian numbers. More specifically, we present recurrence relations of high-order Eulerian numbers, row-generating functions, and row sums of higher-order Eulerian triangles. Furthermore, we investigate the higher-order Eulerian fraction and its alternative form. Some properties of higher-order Eulerian fractions are expressed using differentiation and integration. We derive the inversion relations between second-order Eulerian numbers and Stirling numbers of the second and first kinds. Finally, we provide exact expressions and a computational method for higher-order Eulerian numbers.<\/jats:p>","DOI":"10.3390\/axioms15010016","type":"journal-article","created":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T23:52:27Z","timestamp":1766706747000},"page":"16","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combinatorial Properties and Values of High-Order Eulerian Numbers"],"prefix":"10.3390","volume":"15","author":[{"given":"Tian-Xiao","family":"He","sequence":"first","affiliation":[{"name":"Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/19300980.1959.12467624","article-title":"Eulerian numbers and polynomials","volume":"32","author":"Carlitz","year":"1958","journal-title":"Math. 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