On Ward Numbers and Increasing Schröder Trees

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 Schröder trees} by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schröder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schröder trees). Weighted generalizations extend these bijections to enriched increasing Schröder trees trees and Schröder trees trees, yielding new links to labeled rooted trees. Finally, we deduce a functional equation for weighted increasing Schröder trees, whose solution using Chen's decomposition leads to a combinatorial interpretation of a Lagrange inversion variant.