Bijections in weakly increasing trees via binary trees

TL;DR

This paper introduces four bijections on weakly increasing trees derived from binary trees, providing combinatorial proofs of symmetries and applications to permutation patterns and statistics.

Contribution

It constructs new bijections on weakly increasing trees by manipulating binary tree structures, unifying and extending previous symmetry results.

Findings

Bijective proofs of symmetries in weakly increasing trees

Non-recursive construction of a bijection on plane trees

Applications to permutation patterns and statistics

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 in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we construct four bijections on weakly increasing trees in the same flavor via switching the role of left child and right child of some specified nodes in their corresponding binary trees. Consequently, bijective proofs of the aforementioned symmetries found by Dong et al. and a non-recursive construction of a bijection on plane trees of Deutsch are provided. Applications of some symmetries in weakly increasing trees to permutation patterns and statistics will also be discussed.