Julia E. Bergner, Cedric Harper, Ryan Keller, Mathilde Rosi-Marshall
We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.
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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 ${2^n}n! \binom{2n}{n}$.
On the enumeration of records of rooted trees and rooted forests
本文将有根Cayley树及有根森林记录的生成函数与Cayley树函数相关联，得到了优美的函数方程并探讨了其结果。
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.
For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.
We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step $k$ and the set of planar rooted trees with $k$ edges.
Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests
本文将词洗牌性质推广到根树与根森林，构造根树上的余积和积以建立洗牌双代数及其对偶双代数，刻画并计数原始树，并利用修正洗牌证明Rota-Baxter代数中的性质。
We enumerate injectively $k$-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.
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.
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.
We describe a bijection between $(k,k)$-Fuss-Schröder paths of type $λ$ and certain rooted plane forests with $n(k+1)+2$ vertices. This yields a recursion which allows us to analytically enumerate the number of large $(k,r)$-Fuss-Schröder paths of type $λ$, solving an open question posed by An, Jung, and Kim. Furthermore, we generalize the concept of $(k,r)$-Fuss-Schröder paths to $(k,S)$-Fuss-Schröder paths, in which $r$ can take any value in a given set $S$, and enumerate these paths as well.
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
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。
In this paper, we develop a new method to produce explicit formulas for the number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $ G=C_{n}(s_1,s_2,\ldots,s_k)$ and $ G=C_{2n}(s_1,s_2,\ldots,s_k,n).$ These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form $f_{G}(n)=p\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the parity of $n$. Finally, we find an asymptotic formula for $f_{G}(n)$ through the Mahler measure of the associated …
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 …
Cocycle weighted infinitesimal bialgebras and pre-Lie algebras on rooted trees
本文将装饰平面有根森林空间构造为加权无穷小双代数，通过引入对偶加权1-余环条件定义了加权$Ω$-余环无穷小幺双代数，并建立了不同余积结构间的同构与前李代数结构。
Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations
本文研究轮图中的根生成森林，给出基于Fibonacci和Lucas数的计数公式，并解决根配置的极值问题。
Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C_{k,m}(n)=\frac{1}{mn+1}{(mn+1)k \choose n}, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers.
In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a closed description for those families. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot nu…
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric function…
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。
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can asso…
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.
Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $σ\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $σ$. For each permutation $σ\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(σ)(k-1)k:=σ(1)\cdotsσ(k-2)(k-1)k$ and $n$-vertex forests avoiding $(σ)k(k-1):=σ(1)\cdotsσ(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define…
We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$. Its generating functions will be studied.
Let $\mathcal{D}_0:= \{x + iy \ \vert x, y >0\}$, and let $(L, R)$ be a pair of Möbius transformations corresponding to $\mathrm{SL}_2(\mathbb{N}_0)$ matrices such that $R(\mathcal{D}_0)$ and $L(\mathcal{D}_0)$ are disjoint. Given such a pair (called a left-right pair), we can construct a directed graph $\mathcal{F}(L, R)$ with vertices $\mathcal{D}_0$ and edges $\{(z, R(z))\}_{z \in \mathcal{D}_0} \cup \{(z, L(z))\}_{z \in \mathcal{D}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of $\mathrm{SL}_2(\mathbb{N}_…
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.
Weighted sums of rooted spanning forests on cycles with pendant edges
本文利用矩阵树定理推导出带悬挂边的圈图的有根生成森林加权和的两个公式，并指出其中一个公式可表示为 Chebyshev 多项式的变量变换。
In this paper we present a CAT generation algorithm for Dyck paths with a fixed length n. It is the formalization of a method for the exhaustive generation of this kind of paths which can be described by means of two equivalent strategies. The former is described by a rooted tree, the latter lists the paths by means of three operations which, as we are going to see, are equivalent to visit the tree. These constructions are strictly connected with ECO method and can be encoded by a rule, very similar to the succession rule in ECO, with a finite number of labels for each n. Moreover with a sligh…
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。
The number of rooted spanning forests of bicirculant graphs
本文利用切比雪夫多项式，推导了双循环图的根生成森林数目的闭合公式，并研究了其渐进行为。
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。
Generalisations of multiple zeta values to rooted forests
该研究证明收敛的洗牌树化zeta值有级数表示，提出多重zeta值到有根森林的新推广并研究其代数性质，推导相关经典结果的初等证明，刻画锥zeta值与树化zeta值的关系并给出显式求值公式。
By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota-Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota-Baxter algebra with a bialgebra structure and further a Hopf algebra structure.
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.
Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes.
In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case.
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gröbner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to…
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.
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the conc…
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.
Fano Bott manifolds bijectively correspond to signed rooted forests with some equivalence relation. Using this bijective correspondence, we enumerate the isomorphism classes of Fano Bott manifolds and the diffeomorphism classes of indecomposable Fano Bott manifolds. We also observe that the signed rooted forests with the equivalence relation bijectively correspond to rooted triangular cacti.
Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion
研究离散化平面表面的图拉普拉斯矩阵行列式渐近展开，关联生成树数量和带循环根生成森林的渐近展开与正则化决定量。
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove real-rootedness.` Recently I. Beaton and J. Brown (2020) proved the for almost all graphs the coefficients of the domination polynomial form a unimodal sequence, and C. Barton, J. Brown and D. Pike (2020) proved that the forest polynomial (aka acyclic polynomial) is real-rooted iff $G$ is a forest. Let $\mathcal{A}$ be a graph property, and let $a_i(G)$ be the num…
Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph $G$ and the dimer model defined on a decorated version $\GD$ of this graph \cite{Fisher}. In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristi…
Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles
本文提出了一种广义Connes-Kreimer霍普夫代数，通过加权余圈条件扩展了根树结构，并证明其作为自由对象的性质。
We continue our reformulation of free dendriform algebras, dealing this time with the free dendriform trialgebra generated be Y over planar rooted trees. We propose a 'deformation' of a vectorial coding used in Part I, giving a LL-lattice on rooted planar trees according to the terminology of A. Blass and B. E. Sagan. The three main operations on trees become explicit, giving thus a complementary approach to a very recent work of P. palacios and M. Ronco. Our parenthesis framework allows a more tractable reformulation to explore the properties of the underlying lattice describing operations an…
The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous-Broder Theorem
该研究提出有根森林的序质量恒等式，以此直接证明无需可逆性等假设的扩展Aldous--Broder定理，并在Lean4中形式化相关内容，为图上随机生成森林的采样提供支撑。
Matrix-Forest Theorems
本文推广矩阵-树定理，证明矩阵 $I+L(G)$ 的行列式及余子式分别计数生成有根森林和指定同根关系的森林，并扩展至加权多重图与有向图。
We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth's rotation correspondence between planar binary trees and planar rooted trees. The operadic counterpart of the new bijection is explained. Related to this, the space of planar reduced forests is endowed with a combinatorial Hopf algebra structure. The corresponding structure on the space of planar rooted hyperforests is also described.
Quickly excluding an apex-forest
本文证明了排除apex-forest作为minor的图的layered pathwidth上界，改进了前人结果，并提供了关于rooted minor、treedepth和treewidth的结构性工具及Erdős-Pósa性质的讨论。
For a graph G, let f_{ij} be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The {\em doubly stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^{-1}=I+L, where L is the Laplacian matrix of G. We show that fo…
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。
We propose the reliability constrained k-rooted minimum spanning forest, a relevant optimization problem whose aim is to find a k-rooted minimum cost forest that connects given customers to a number of supply vertices, in such a way that a minimum required reliability on each path between a customer and a supply vertex is satisfied and the cost is a minimum. The reliability of an edge is the probability that no failure occurs on that edge, whereas the reliability of a path is the product of the reliabilities of the edges in such path. The problem has relevant applications in the design of netw…
Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph
本文分类了 Grossman-Larson Hopf 代数在根树基上的非负实特征，揭示其由 Kingman paintbox 参数化，并刻画了 Hoffman 根树图上的中心测度面及 Martin 边界结构。
Regular packing of rooted hyperforests with root constraints in hypergraphs
本文研究无向图与超图中带根约束的森林正则打包问题，刻画了k个森林打包满足顶点出现次数、各森林连通分量数及总连通分量数约束的存在性条件，并推广了Bérczi和Frank的定理。
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.
For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path, $A_F(q)$ specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of $A_F(q)$ is unimodal and that if $\{T_{n}\}$ is a sequence of trees with $|T_{n}| = n$ and maximal down degree $D_{n} = O(n^{0.5-ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.
We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(ε)=(I+εL)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and εis a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to …
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