Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers

Abstract

The $n^{\text{th}}$ small Schröder number is $s(n)={\sum }_{k\ge 0}s(n,k)$, where $s(n,k)$ denotes the number of plane rooted trees with $n$ leaves and $k$ internal nodes that each has at least two children. In this manuscript, we focus on the weighted small Schröder numbers $s_d(n)={\sum }_{k\ge 0}s(n,k)d^k$, where $d$ is an arbitrary fixed real number. We provide recursive and asymptotic formulas for $s_d(n)$, as well as some identities and combinatorial interpretations for these numbers. We also establish connections between $s_d(n)$ and several families of Dyck paths.

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• Operads from posets and Koszul duality
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