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.
Full version: pdf,
(Concerned with sequences
Received May 2 2020; revised version received December 23 2020.
Published in Journal of Integer Sequences,
December 28 2020.
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