Journal of Integer Sequences, Vol. 24 (2021), Article 21.1.1

Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers

Yu Hin (Gary) Au
Department of Mathematics and Statistics
University of Saskatchewan
Saskatoon, SK S7N 5E6


The nth small Schröder number is s(n) = Σk≥0 s(n, k), where s(n, k) denotes the number of plane rooted trees with n leaves and k internal nodes, each having at least two children. In this manuscript, we focus on the weighted small Schröder numbers sd(n) = Σk≥0 s(n,k)dk, where d is an arbitrary fixed real number. We provide recursive and asymptotic formulas for sd(n), as well as some identities and combinatorial interpretations for these numbers. We also establish connections between sd(n) and several families of Dyck paths.

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(Concerned with sequences A000108 A001003 A006318 A071684 A071688 A086810 A090181 A107841 A131763 A131765.)

Received May 2 2020; revised version received December 23 2020. Published in Journal of Integer Sequences, December 28 2020.

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