Some Properties and Combinatorial Implications of Weighted Small Schr\"oder Numbers

TL;DR

This paper explores weighted small Schr"oder numbers, providing recursive, asymptotic formulas, identities, combinatorial interpretations, and connections to Dyck paths, thereby deepening understanding of their properties and implications.

Contribution

It introduces and analyzes weighted small Schr"oder numbers, offering new formulas, identities, and combinatorial insights that extend classical understanding of these numbers.

Findings

Derived recursive formulas for weighted small Schr"oder numbers

Established asymptotic behavior of these numbers

Connected weighted numbers to Dyck path families

Abstract

The $n^{\text{th}}$ small Schr\"oder number is $s(n) = \sum_{k \geq 0} s(n,k)$, 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 Schr\"oder numbers $s_d(n) = \sum_{k \geq 0} s(n,k) d^k$, 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.