More bijective combinatorics of weakly increasing trees

Abstract

As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin–Ma–Ma–Zhou (2021). Various intriguing connections and bijections for weakly increasing trees have already been found and the purpose of this paper is to present yet more bijective combinatorics on this unified object. Two of our main contributions are

• •

  extension of an equidistribution result on plane trees due to Eu–Seo–Shin (2017), regarding levels and degrees of nodes, to weakly increasing trees;

• •

  a new interpretation of the multiset Schett polynomials in terms of odd left/right chains on weakly increasing binary trees.

Interesting consequences are discussed, including new tree interpretations for the Jacobi elliptic functions and Euler numbers. Relevant enumerative results are also presented, involving recurrence relations, exponential generating functions and context-free grammars.