Weakly increasing trees on a multiset

doi:10.1016/j.aam.2021.102206

Advances in Applied Mathematics

Volume 129, August 2021, 102206

https://doi.org/10.1016/j.aam.2021.102206 Get rights and content

Highlights

•
The concept of weakly increasing trees on a multiset is introduced.
•
For a general multiset M, a compact product formula for the number of weakly increasing trees on M is derived.
•
The M-Eulerian-Narayana polynomial for weakly increasing trees on M is defined.
•
It is proved combinatorially that the M-Eulerian-Narayana polynomial is γ-positive.

Abstract

In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set ${0, 1, \dots, n}$ . We define the M-Eulerian–Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing trees on a general multiset. Inspired by some remarkable equidistributions between multipermutations and s-inversion sequences, we establish two connections between our M-Eulerian–Narayana polynomials for the multiset $M = {1^{2}, 2^{2}, \dots, n^{2}}$ (resp. $M = {1^{2}, 2^{2}, \dots, {(n - 1)}^{2}, n}$ ) and Savage and Schuster's s-Eulerian polynomials for the sequence $s = (1, 1, 3, 2, 5, 3, \dots, 2 n - 1, n, n + 1)$ (resp. $s = (1, 1, 3, 2, 5, 3, \dots, 2 n - 1, n)$ ). We also derive equidistributions that involve some natural tree statistics among weakly increasing trees on different multisets. Via introducing a group action on weakly increasing trees, we prove combinatorially the γ-positivity of the M-Eulerian–Narayana polynomials for a general multiset M. As an application of this γ-positivity result, we obtain a combinatorial interpretation of γ-coefficients of descent polynomials of permutations on the multiset ${1^{2}, 2^{2}, \dots, n^{2}}$ in terms of weakly increasing trees.

Introduction

Let n be a positive integer and $[n] : = {1, 2, \dots, n}$ . A labeled tree on $[n]$ is a rooted tree comprising $n + 1$ nodes that are labeled by distinct integers in the set ${0, 1, 2, \dots, n}$ . An increasing tree on $[n]$ is a labeled tree in which the labels of the nodes are increasing along any path from the root. Note that 0 is the root of any increasing tree.

There are many close connections between increasing trees and some other fundamental combinatorial structures, such as permutations, inversion sequences and increasing binary trees. Please refer to [18], [22], [25], [29], [40], [49]. Some systematic and fundamental studies of increasing tree families are given in [3], [19], [42].

For any increasing tree T, denote by $leaf (T)$ and $int (T)$ the number of leaves and internal nodes in T, respectively. Let $T_{n}$ be the set of increasing trees on $[n]$ . Then the Eulerian polynomial $A_{n} (x)$ is given as follows: $A_{n} (x) : = \sum_{T \in T_{n}} x^{int (T) - 1} = \sum_{T \in T_{n}} x^{leaf (T) - 1} .$ The Eulerian polynomial has been extensively studied (cf. [15], [20], [23], [26]) from different aspects, frequently using the interpretation as the descent or ascent polynomials over permutations: $A_{n} (x) = \sum_{π \in S_{n}} x^{des (π)} = \sum_{π \in S_{n}} x^{asc (π)},$ where $S_{n}$ is the set of permutations of $[n]$ and $des (π)$ (resp. $asc (π)$ ) is the number of descents (resp. ascents) of π. The reader is referred to the book [49, p. 25] of Stanley for a natural bijection $ψ : S_{n} \to T_{n}$ that transforms the statistic $des$ to $leaf$ .

For any $1 \leq k \leq n$ , the Narayana numbers $N (n, k)$ are defined as $N (n, k) = \frac{1}{n} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} n \\ k - 1 \end{matrix}),$ which are listed as sequence A001263 in [39]; see also [17], [30], [50], [51], [54]. The Narayana polynomials are defined by $N_{n} (x) = \sum_{k = 0}^{n - 1} N (n, k) x^{k}, n \geq 1 .$ These polynomials were studied extensively in the literature [5], [12], [14], [52], [36].

A plane tree P can be defined recursively as follows: a node v is one designated vertex, which is called the root of P. Then either P contains only one node v, or it has a sequence $(P_{1}, P_{2}, \dots, P_{k})$ of subtrees $P_{i}$ , $i \in [k]$ , each of which is a plane tree. So, the subtrees of each node are linearly ordered. We write these subtrees in the order left to right when we draw such trees. We also write the root v on the top and draw an edge from v to the root of each of its subtrees. It is well known [10], [43], [39] that plane trees with n edges and k leaves are enumerated by the Narayana number $N (n, k)$ .

In Subsection 1.1, each node in an increasing tree has a distinct label. In [27], [28], multilabeled tree families, which are extensions of increasing trees, are studied, where the nodes in the tree are equipped with a set or a sequence of labels. In [4], a weaker model of increasing trees, in which repetitions of the labels can appear but the sequence of labels is (strictly) increasing along each branch starting from the root, was introduced.

In this work, we introduce the definition of weakly increasing trees on a multiset, which gives a unified extension of increasing trees and plane trees. Let $(p_{1}, p_{2}, \dots, p_{n})$ be a sequence of positive integers. Let $M = {1^{p_{1}}, 2^{p_{2}}, \dots, n^{p_{n}}}$ be a multiset. We define $M_{0} = M \cup {0}$ and $N = N (M) = \sum_{i = 1}^{n} p_{i}$ .

Definition 1.1

A weakly increasing tree on M is a plane tree such that

(i)
it contains $N + 1$ nodes that are labeled by integers in the multiset $M_{0}$ ;
(ii)
each sequence of labels along a path from the root to any leaf is weakly increasing;
(iii)
labels of the children of each node are in nonincreasing order from left to right.

Denote by

T_{M}

the set of weakly increasing trees on M.

Example 1.2

Take $M = {1^{2}, 2^{2}}$ . We draw all 18 weakly increasing trees on M in Fig. 1.

By Definition 1.1, weakly increasing trees on $M = [n]$ are exactly increasing trees on $[n]$ , while weakly increasing trees on $M = {1^{n}}$ are exactly plane trees on $n + 1$ nodes.

Let T be a weakly increasing tree in $T_{M}$ . For any $j \in M_{0}$ , if the node j has at least one child in T, then we say that j is an internal node of T; otherwise, j is a leaf of T. Define two multisets $Int (T)$ and $Leaf (T)$ by $Int (T) = {j | j \in M_{0} and the node j is an internal node of T}, Leaf (T) = {j | j \in M_{0} and the node j is a leaf of T} .$ Let $int (T) = | Int (T) |$ and $leaf (T) = | Leaf (T) |$ . Clearly, $Int (T) \cup Leaf (T) = M_{0}$ and $int (T) + leaf (T) = N + 1$ .

Example 1.3

For the 18 weakly increasing trees T in Fig. 1, we list their $Int (T)$ and $int (T)$ in the following table.

Let us introduce the polynomial $A_{M} (x)$ as $A_{M} (x) = \sum_{T \in T_{M}} x^{int (T) - 1} .$ For instance, if $M = {1^{2}, 2^{2}}$ , then $A_{M} (x) = 1 + 8 x + 8 x^{2} + x^{3}$ . By Definition 1.1, when $M = [n]$ , $A_{M} (x) = A_{n} (x)$ is the nth Eulerian polynomial; when $M = {1^{n}}$ , $A_{M} (x) = N_{n} (x)$ is the nth Narayana polynomial. Hence, we call $A_{M} (x)$ the M-Eulerian–Narayana polynomial for the multiset M.

Section snippets

Definitions and main results

In this section, we introduce further motivations of this work and state our main results whose proofs will be given in subsequent sections.

Proof of Theorem 2.1

We need a combinatorial identity before the proof of Theorem 2.1.

Lemma 3.1

For any $a \geq 0$ and $p \geq 1$ , we have $\sum_{k = 1}^{p} \frac{k}{p} (\begin{matrix} 2 p - 1 - k \\ p - 1 \end{matrix}) (\begin{matrix} a + k \\ k \end{matrix}) = \frac{a + 1}{a + 2 p + 1} (\begin{matrix} a + 2 p + 1 \\ p \end{matrix}) .$

Proof

We first write the left-hand side of (8) as $\frac{1}{p} \sum_{k = 1}^{p} k (\begin{matrix} a + k \\ k \end{matrix}) (\begin{matrix} p - 1 + (p - k) \\ p - k \end{matrix}) .$ Since $\sum_{k \geq 0} (\begin{matrix} n + k \\ k \end{matrix}) x^{k} = {(1 - x)}^{- (n + 1)}$ , we have $\sum_{k \geq 1} k (\begin{matrix} a + k \\ k \end{matrix}) x^{k} = (a + 1) x {(1 - x)}^{- (a + 2)} and \sum_{k \geq 0} (\begin{matrix} p - 1 + k \\ k \end{matrix}) x^{k} = {(1 - x)}^{- p} .$ Thus, the summation in (9) is the coefficient of $x^{p}$ in $\frac{a + 1}{p} x {(1 - x)}^{- (a + p + 2)},$ which equals the right-hand side of (8). □

We are ready for the proof of Theorem 2.1.

Proof of Theorem 2.1

Let $M^{'} = {1^{p_{1}}, 2^{p_{2}}, \dots, {(n - 1)}^{p_{n - 1}}}$ and $[$

Proofs of Theorems 2.5 and 2.7

Before we start to prove Theorem 2.5, we need to prepare several lemmas. Let s be a sequence of positive integers. An s-lecture hall partition is a finite integer sequence $λ = (λ_{1}, λ_{1}, \dots λ_{n})$ satisfying $0 \leq \frac{λ_{1}}{s_{1}} \leq \frac{λ_{2}}{s_{2}} \leq \dots \leq \frac{λ_{n}}{s_{n}} .$ Let $L_{n}^{(s)}$ denote the set of all s-lecture hall partitions of length n.

Lemma 4.1

Savage and Schuster [44]

Let $s = (s_{1}, s_{2}, \dots, s_{n}, \dots)$ be a sequence of positive integers and $L_{n}^{(s; j, i)} = | {λ \in L_{n}^{s} | λ_{n} \leq j s_{n} + i} |$ .

(i)
$L_{n}^{(s; j, i)}$ satisfies the recurrence relation $L_{n}^{(s; j, i)} = {\begin{matrix} 1 & if n = 0 or i = j = 0, else, \\ L_{n}^{(s; j - 1, s_{n})} & if i = 0, else, \\ L_{n}^{(s; j, i - 1)} + L_{n - 1}^{(s; j, ⌊ i} \end{matrix}$

Proof of Theorem 2.11

The symmetry (5) in Theorem 2.11 is an immediate consequence of the following recurrence relation for $F_{n; A} (x, y, p, b, c)$ .

Lemma 5.1

Let $n \geq 1$ and $A \subseteq [n]$ . Then

(i)
when $n \in A$ , we have $F_{n; A} (x, y, p, b, c) = \frac{\partial (x y F_{n - 1; B} (x, y, p, b, c))}{\partial x} + \frac{\partial (x y F_{n - 1; B} (x, y, p, b, c))}{\partial y},$ where $B = A ∖ {n}$ ;
(ii)
when $n \notin A$ , we have $F_{n; A} (x, y, p, b, c) = (b y + p x) F_{n; B^{'}} (x, y, p, b, c) + \frac{c x y}{2} (\frac{\partial (F_{n; B^{'}} (x, y, p, b, c))}{\partial x} + \frac{\partial (F_{n; B^{'}} (x, y, p, b, c))}{\partial y})$ with $F_{1; 1} (x, y, p, b, c) = 1$ , where $B^{'} = A \cup {n}$ .

Proof

First we consider the case $n \in A$ . For any $T \in T_{n - 1; B}$ , we add a node n to be a child of a node u in T and denote by $T^{'}$

A bijection with multipermutations

In this section, we generalize the classical bijection $ψ : S_{n} \to T_{n}$ in [49, p. 25] to one between restricted weakly increasing trees and permutations of a multiset.

For a multiset M, let $S_{M}$ be the set of permutations of M. Let T be a weakly increasing tree in $T_{M}$ . We construct a permutation $σ_{T} \in S_{M}$ recursively by the following steps:

•
Step 1. If T contains only one node, then let $σ_{T} = \emptyset$ .
•
Step 2. Otherwise, suppose that the children of the root 0 are $i_{1}, i_{2}, \dots, i_{k}$ from left to right, where $i_{1} \geq i_{2} \geq \dots \geq i_{k}$ . Deleting

Generalized Foata–Strehl action on weakly increasing trees

In this section, we develop a generalized Foata–Strehl group action on weakly increasing trees which leads to the proofs of Theorem 2.12, Theorem 2.19.

In order to define our group action, we need first to introduce an order for the nodes of a rooted ordered tree (labeled or unlabeled). For such a tree, the order start at the root node and goes as far as it can down through the rightmost unexplored branch, then backtracks until another unexplored (rightmost) branch was found, and then explores

Acknowledgments

The first author was supported by the National Natural Science Foundation of China grant 11871247 and the project of Qilu Young Scholars of Shandong University. The second author was supported by the National Natural Science Foundation of China grant 11571235. The third author was supported by the National Natural Science Foundation of China grants 11401083 and 12071063.

The authors thank Tongyuan Zhao for telling us that the formula (1) could be further simplified and the referee for his

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Citation Excerpt :
Lin–Ma–Ma–Zhou [15] Weakly increasing trees were introduced by Lin et al. [15] as unified generalization of plane trees and increasing trees. Recall that a plane tree is a rooted tree in which the children of each node are linearly ordered.
Let $A_{n}^{(p)} (t)$ (resp. $B_{n}^{(p)} (t)$ ) be the descent polynomial on permutations of the multiset ${1^{p}, 2^{p}, \dots, n^{p}}$ (resp. ${1^{p}, 2^{p}, \dots, n^{p}, n + 1}$ ). The $γ$ -positivity of $A_{n}^{(p)} (t)$ was known but to give a combinatorial interpretation for the corresponding $γ$ -coefficients still remains open. In this paper we
• manage to find a combinatorial interpretation for the $γ$ -coefficients of $A_{n}^{(p)} (t)$ via the model of weakly increasing trees;
• and prove that $B_{n}^{(p)} (t)$ has bi- $γ$ -positivity expansion $B_{n}^{(p)} (t) = a_{n} (t) + t b_{n} (t),$ where $b_{n} (t) = (p - 1) A_{n}^{(p)} (t)$ .
The first result, whose proof makes use of Chen’s general bijective algorithm for trees and a new decomposition of weakly increasing trees, answers a recent open problem posed by Lin–Ma–Ma–Zhou. The latter result, which we provide both a computational proof and a short combinatorial proof, extends a bi- $γ$ -positivity result due to Ma–Ma–Yeh from $p = 2$ to general $p$ . Their proof for $p = 2$ uses Chen’s context-free grammar and seems hard to be adopted for general $p$ .
Two bijections on weakly increasing trees
2022, Discrete Mathematics
Weakly increasing tree is a new kind of multiset-labeled tree introduced in a recent work of Lin-Ma-Ma-Zhou [9], which naturally unifies the classical concepts of plane trees and increasing trees on the set ${1, 2, \dots, n}$ . Two bijections defined on weakly increasing trees are introduced. The first map has its roots in a bijection on plane trees due to Deutsch [4]. The second map is a closely related variant and turns out to be an involution. This involution amounts to give a combinatorial proof of certain equidistribution results for a sextuple of tree statistics over weakly increasing trees defined on any multiset M, extending a previous result of Lin-Ma-Ma-Zhou on restricted M.

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Weakly increasing trees on a multiset

Highlights

Abstract

Introduction

Section snippets

Definitions and main results

Proof of Theorem 2.1

Proofs of Theorems 2.5 and 2.7

Savage and Schuster [44]

Proof of Theorem 2.11

A bijection with multipermutations

Generalized Foata–Strehl action on weakly increasing trees

Acknowledgments

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J. Comb. Theory, Ser. A

Eur. J. Comb.

Discrete Math.

J. Comb. Theory, Ser. A

J. Comb. Theory, Ser. A

J. Comb. Theory, Ser. A

J. Comb. Theory, Ser. A

J. Comb. Theory, Ser. A

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