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 Refined enumeration of vertices among all rooted ordered $d$-trees Bull. Korean Math. Soc.Published online July 9, 2019 Sangwook Kim, Seunghyun Seo, and Heesung Shin Chonnam National University, Kangwon National University, Inha University Abstract : In this paper, we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature. Keywords : d-tree, bijection, enumeration MSC numbers : 05A10, 05A15, 05C05 Full-Text :