Bijections in weakly increasing trees via binary trees

TL;DR

Let $M$ be a multiset and $\mathcal{T}_M$ the set of weakly increasing trees on $M$; the associated weakly increasing binary trees $\mathcal{B}_M$ via the map $\rho$ facilitate three involutions that realize the symmetries studied by Dong et al. on plane trees, and extend to a unified group-action viewpoint. The paper also introduces a generating-function framework with refined Narayana/Motzkin-type polynomials $N_n(u_1,u_2,u_3;v_1,v_2)$ counting five statistics, and provides a non-recursive construction of Deutsch's bijection in the binary-tree setting. Applications to permutation patterns include refined peak statistics and two symmetries in $312$-avoiding permutations, including involutions $\Lambda$ and $\Upsilon$ that map $DES$ to $ASC$-type statistics and relate consecutive patterns $1324$ and $3241$. Overall, the methods reveal the central role of binary-tree switches as a unifying tool for bijections between weakly increasing trees, plane trees, and permutation classes, with concrete generating-function and pattern-avoidance consequences.

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.

Figures (10)

• Figure 1: A weakly increasing tree on $\{1^2,2^4,3^3,4^2\}$ and its corresponding weakly increasing binary tree under the bijection $\rho$.
• Figure 2: An example of the involution $\Phi=\rho^{-1}\circ\phi\circ\rho$.
• Figure 3: An example of the involution $\psi$.
• Figure 4: An example of constructing a tip-augmented plane tree.
• Figure 5: 3-node binary trees

Theorems & Definitions (49)

• Definition 1.1: Weakly increasing trees Lin20
• Theorem 1.2
• Definition 1.3: Tip-augmented weakly increasing trees
• Theorem 1.4
• Corollary 1.5: Dong-Du-Ji-Zhang Dong1
• Proposition 1.6: Dong-Du-Ji-Zhang Dong1
• Theorem 1.7
• Definition 2.1: Weakly increasing binary trees LM
• Lemma 2.2
• proof