On Ward Numbers and Increasing Schr\"oder Trees

TL;DR

This paper establishes a combinatorial link between Ward numbers, set partitions, and increasing Schr"oder trees, providing new bijections, weighted generalizations, and a functional equation with a Lagrange inversion interpretation.

Contribution

It proves Ward numbers count increasing Schr"oder trees, constructs a direct bijection, and extends these results to weighted and enriched trees with new combinatorial insights.

Findings

Ward numbers count increasing Schr"oder trees.

A direct bijection between total partition trees and increasing Schr"oder trees is constructed.

A functional equation for weighted increasing Schr"oder trees is derived and interpreted via Lagrange inversion.

Abstract

The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schr\"oder trees} by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schr\"oder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schr\"oder trees). Weighted generalizations extend these bijections to enriched increasing Schr\"oder trees trees and Schr\"oder trees trees, yielding new links to labeled rooted trees. Finally, we deduce a functional equation for weighted increasing Schr\"oder trees, whose solution using Chen's decomposition leads to a combinatorial interpretation of a Lagrange inversion variant.