David Callan
Lajos Takacs gave a somewhat formidable alternating sum formula for the number of forests of unrooted trees on $n$ labeled vertices. Here we use a weight-reversing involution on suitable tree configurations to give a combinatorial derivation of Takacs' formula.
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A signed labeled forest is defined as a (plane) forest labeled by {1,2,..., n} along with minus signs associated to some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number, the flag major index and the R-major index on signed labeled forests. They can be considered as type B analogues of the indices for labeled forests introduced by Bjorner and Wachs. The flag major index for signed labeled forests is based on the flag major index on signed permutations introduced by Adin and Roichman, whereas the R-major index for signed label…
In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes and multinomial identities are given.
Björner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed. Our results extend the result of Björner and Wachs and generalize results for permutations. We also introduce analogous statistics for signed label…
Stanley-Wilf Limits for Patterns in Rooted Labeled Forests
该研究基于Garg和Peng的工作，用解析方法证明了有根标号森林中规避单模式及部分模式集合时的Stanley-Wilf猜想类比结论，可计算对应极限的收敛下界，并提出了相关开放研究问题。
The theme of this article is the algebraic combinatorics of leaf-labeled rooted binary trees and forests of such trees. The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests.
We present an algorithmic mapping from permutations of length dn to labeled n-node d-ary trees and back again. Given such a bijection, one can interpret each of the factorials in the formula for the Catalan numbers as a count of permutations.
Wilf Equivalences for Patterns in Rooted Labeled Forests
该研究延续有根森林模式规避研究，通过推广森林Simion-Schmidt双射完成特定模式集的森林-Wilf等价分类，借助森林版Goulden-Jackson簇方法发现新的非平凡c-森林-Wilf等价族，还证明超强c-森林-Wilf等价在森林中是平凡的。
Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
本文研究了枚举有根带标号树森林的生成多项式矩阵，证明了其系数全正性与Hankel全正性，并推广至通用有根森林多项式，利用生产矩阵和指数Riordan数组完成证明。
We consider conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. In particular, we determine nilpotent matrices whose orbit under symmetric group afford the sign representation. Applications to rook theory are offered.
Distribution of new statistics of parking functions and their generalizations
研究停车函数与标号森林的计数，引入新统计量并通过双射推广，确定其联合分布，解释并显式构造 Stanley–Yin 等分布双射。
This paper discusses the enumeration for the total number of all rooted spanning forests of the labeled complete tripartite graph. We enumerate the total number by a combinatorial decomposition.
Exceptional sequences and rooted labeled forests
本文通过表示论建立了有根标记森林与直线定向 $A_n$ 型箭图完全例外序列之间的双射，并研究了扩展辫群及 Garside 元素在这些对象上的自然作用及其与丛理论的联系。
A loop-augmented forest is a labeled rooted forest with loops on some of its roots. By exploiting an interplay between nilpotent partial functions and labeled rooted forests, we investigate the permutation action of the symmetric group on loop-augmented forests. Furthermore, we describe an extension of the Foulkes' conjecture and prove a special case. Among other important outcomes of our analysis are a complete description of the stabilizer subgroup of an idempotent in the semigroup of partial transformations and a generalization of the (Knuth-Sagan) hook length formula.
We say that an unordered rooted labeled forest avoids the pattern $π\in\mathcal{S}_n$ if the sequence obtained from the labels along the path from the root to any vertex does not contain a subsequence that is in the same relative order as $π$. We enumerate several classes of forests that avoid certain sets of permutations, including the set of unimodal forests, via bijections with set partitions with certain properties. We also define and investigate an analog of Wilf-equivalence for forests.
For every labeled forest $\mathsf{F}$ with set of vertices $[n]$ we can consider the subgroup $G$ of the symmetric group $S_n$ that is generated by all the cycles determined by all maximal paths of $\mathsf{F}$. We say that $G$ is the chain group of the forest $\mathsf{F}$. In this paper we study the relation between a forest and its chain group. In particular, we find the chain groups of the members of several families of forests. Finally, we prove that no copy of the dihedral group of cardinality $2n$ inside $S_n$ can be achieved as the chain group of any forest.
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.
We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher label. This solves a problem posed by Shin and Zeng in a recent article. We also provide a generalization of this identity that translates to a formula for the number of rooted spanning forests with given indegree sequence.
Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs
本文研究了元素为 $t_{n,k} = \binom{n}{k} n^{n-k}$ 的下三角矩阵的组合模型，证明其全正性及行生成多项式序列的 Hankel 全正性，并利用生产矩阵与指数 Riordan 阵将其推广至无穷多不定元的多项式。
Let $A$ be a commutative $k$-algebra over a field of $k$ and $Ξ$ a linear operator defined on $A$. We define a family of $A$-valued invariants $Ψ$ for finite rooted forests by a recurrent algorithm using the operator $Ξ$ and show that the invariant $Ψ$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $α(T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in \bT_n} \frac 1{α(T)} Ψ(T)$, where $\bT_n$ is the set of rooted trees …
In 1966, Cummins introduced the "tree graph": the tree graph $\mathbf{T}(G)$ of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge, i.e., two spanning trees $T_1$ and $T_2$ are adjacent if $T_2 = T_1 -e +f$ for some edges $e\in T_1$ and $f\notin T_1$. The tree graph of a connected graph need not be connected. To obviate this difficulty we define the "forest graph": let $G$ be a labeled graph of order $α$, finite or infinite, and let $\mathfrak{N}(G)$ be the set of all labeled maximal fo…
Recently Han obtained a general formula for the weight function corresponding to the expansion of a generating function in terms of hook lengths of binary trees. In this paper, we present formulas for k-ary trees, plane trees, plane forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.
In this paper, we give a bijection between rooted labeled ordered forests with a selected subset of their leaves and the regions of the type $C$ Catalan arrangement in $\R^n$. We thus obtain a bijective proof of the well-known enumeration formula of these regions ${2^n}n! \binom{2n}{n}$.
Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1, \ldots, n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit di…
We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this work: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Se…
In this paper, we construct explicitly a noncommutative symmetric (${\mathcal N}$CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the ${\mathcal N}$CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficient…
Enumerative combinatorics of unlabeled and labeled time-consistent galled trees
研究无标号与有标号时间一致瘿树的计数问题，用符号方法推导计数公式，并给出生成函数与渐近结果。
Combinatorics on bi-$γ$-positivity of $1/k$-Eulerian polynomials
本文通过有序标记森林模型，为$1/k$-欧拉多项式的双$\gamma$-系数提供了组合解释，方法包括构造双射、引入广义Foata--Strehl作用及两个关键变换。
We introduce the generic Lah polynomials $L_{n,k}(ϕ)$, which enumerate unordered forests of increasing ordered trees with a weight $ϕ_i$ for each vertex with $i$ children. We show that, if the weight sequence $ϕ$ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials $L_n(ϕ,y)$ is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive typ…
Quota Trees
该研究在有向图中引入配额树概念，给出了其存在的充要条件，解决了 Myhill-Nerode 类大小问题，并推广了矩阵树定理与 Edmonds 算法。
All-Minors Matrix-Tree Theory for Superport Networks: Completed Quotient-Incidence Determinants and Conductance-Weighted Subdivision Extensions
本文针对超端口网络的缺失全子式情况，确定了生成森林的符号规则，通过完备商关联行列式给出任意响应子式的计算式，简化了雅可比因子，明确了非零条件。
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