The lattice dimension of a tree
Sergei Ovchinnikov
图G的格维数是G可以等距嵌入的立方格的最小维数。证明了n叶树的格维数为$\lceil n/2 \rceil$。
The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. We prove that the lattice dimension of a tree with n leaves is $\lceil n/2 \rceil$.
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We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.
Supersolvable LL-lattices of binary trees
证明了某些二叉树偏序集是超可解格，并给出了显式EL-标号。计算它们的特征多项式，以不同的方式恢复它们的已知因式分解。
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous grammars that allow to generate tree-like structures and obtain fixed-point functional equations to enumerate these. We also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the Tamari lattice. Finally, we investigate other families of binary trees that are also close…
We introduce a new combinatorial structure: the metasylvester lattice on decreasing trees. It appears in the context of the $m$-Tamari lattices and other related $m$-generalizations. The metasylvester congruence has been recently introduced by Novelli and Thibon. We show that it defines a sublattice of the $m$-permutations where elements can be represented by decreasing labelled trees: the metasylvester lattice. We study the combinatorial properties of this new structure. In particular, we give different realizations of the lattice. The $m$-Tamari lattice is by definition a sublattice of our n…
The existence of greatest lower bounds in the imbalance order of path-length sequences of binary trees is seen to be a consequence of a joint monotonicity property of the greater and lower expension operations. Path length sequences that are join-irreducible in the imbalance lattice are characterized.
Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice pa…
The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can …
The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can …
Balanced Spanning Trees for Triangular Strip Lattices
本文针对三角条带格，通过递推关系和渐近分析，研究了平衡生成树的计数及其比例，并给出了均匀随机选取时平衡生成树的极限概率。
A bijection is constructed between two sets of height restricted lattice paths by means of translating them in two tree classe, namely plane trees and Elena trees. An old bijection between them can be used now for that actual problem.
We give new interpretations of the $ν$-Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of $ν$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of $ν$-trees and componentwise order, we provide a simple description of the lattice property. We also show that the $ν$-Tamari lattice is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey's lattice conjecture concerning the …
On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games
本文研究阶乘与第一类斯特林数的交错和，给出指数生成函数并证明其非负性，将其与递增树博弈、弱序到Tamari格的商及平面树双射相联系，并给出有根树博弈胜负的等价刻画。
In the hyperbolic plane there are infinite regular lattices. From a fix vertex of a lattice tree graphs can be constructed recursively to the next layers with edges of the lattice. In this article we examine the properties of the growing of trees and the probabilities of length of trees considering the vertices on level i.
Heinz Huber (1956) considered the following problem on the the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree.
The lattice polynomials $L_{i,j}(x)$ are introduced by Hough and Shapiro as a weighted count of certain lattice paths from the origin to the point $(i,j)$. In particular, $L_{2n, n}(x)$ reduces to the generating function of the numbers $T_{n,k}={1\over n}{n-1+k\choose n-1}{2n-k\choose n+1}$, which can be viewed as a refinement of the $3$-Catalan numbers $T_n=\frac{1}{2n+1}{3n\choose n}$. In this paper, we establish a correspondence between $12312$-avoiding partial matchings and lattice paths, and we show that the weighted count of such partial matchings with respect to the number of crossings …
In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d'algèbres de Hopf combinatoires à partir d'une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous metto…
In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of $d$-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the number of spanning trees by the well-known matrix tree theorem. Asymptotics for $2$ and $3$ component rooted spanning forests in these graphs are also derived. Moreover, we express the number of spanning trees in a $2$-dimensional square lattice in terms of the one in a $2$-dimensional discrete torus and also in the quartered Aztec diamond. As a consequence, we…
We introduce new combinatorial objects, the interval- posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear operator on each tree T counts the number of trees smaller than T in the Tamari order. Then we show that a similar m + 1-linear operator is also used in the functionnal equation of m-Tamari intervals. We explain how the m-Tamari …
Geometric bounds for spanning tree entropy of planar lattice graphs
本文利用双曲交错链、直角双曲多面体及正则理想双锥的体积，证明了平面格图生成树熵尖锐上下界猜想的无穷多种情形，并给出易于计算且精度优异的数值估计。
Trees and Matchings
在这篇文章中，Temperley的双射之间的生成树的正方形网格的一方面，和完美匹配（也称为二聚体覆盖）的正方形网格的另一方面，被扩展到设置一般平面有向（和无向）图，其中的边缘进行非负的权重，诱导一组的生成树的权重。我们证明了任何可平面图G的加权有向生成树（通常称为树状图）可以与相关可平面图H的完美匹配一一保持权对应.这个结果的一个特例是六边形蜂窝格的完美匹配和三角格的有向生成树之间的双射。另一种特殊情况给出了“正方形-八边形”格的完美匹配和有向加权生成树在有向加权版本的Carnival格上的对应关系。结合Kenyon的结果，我们的主要定理允许我们计算任意（有向，加权）可平面图上随机生成树的所有圆柱事件的测度。相反，在完美匹配模型来自树模型的情况下，Wilson算法允许我们快速生成完美匹配的随机样本。
Some Exact Results for Spanning Trees on Lattices
对于$n$ -顶点，$d$ -维格$Λ$和$d \ge 2$，生成树$N_{ST}(Λ)$的数目在热力学极限下以$\exp(n z_Λ)$的形式渐近增长.我们给出了$d$维体心立方晶格上生成树的渐近增长常数$z_{bcc(d)}$的一个精确的封闭形式结果。我们还给出了面心立方晶格上$z_{fcc}$的精确积分表达式和$4 \cdot 8 \cdot 8$晶格上$z_{488}$的精确闭合表达式。
We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order, Tamari, Cambrian and boolean lattices and the classical maps between them. On the geometric side, we provide both the vertex and facet descriptions of a polytope realizing the rotation lattice, specializing to the permutahedron, the associahedra, and certain graphical zonotopes. On the algebraic side, we construct a Hopf algebra on permutrees containing the …
Spanning Trees on Lattices and Integration Identities
对于具有$n$顶点和维度$d$等于或大于2的格$Λ$，生成树$N_{ST}(Λ)$的数量在热力学极限下渐近地增长为$\exp(n z_Λ)$。我们提出了几个格上的生成树的渐近增长常数$z_Λ$的精确积分表达式。在计算中采用不同的单元格，可以得到许多积分恒等式。我们还给出了$p$顶点插入每条边的$k$ -正则格的同胚展开式的$z_{Λ(p)}$.
Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces
我们考虑格上生成树的枚举问题。给出了尺寸为N_1 × N_2 × N的超立方格的生成树母函数的封闭表达式。x N_d在d维自由，周期，自由和周期的组合边界条件下。结果也得到了一个简单的四次网嵌入两个不可定向的表面，一个Moebius带和克莱因瓶。我们的研究结果是基于使用一个公式表示的生成树生成函数的特征值相关联的树矩阵。给出了该公式的初等推导。
Complexes of trees and nested set complexes
我们展示了一个身份的抽象单纯复形之间的研究复杂的树和简化的最小套集复杂的分划格。我们的结论是，分划格的序复形可以通过一系列的恒星划分从树的复形中得到。我们提供了一个明确的上同调的基础上复杂的树木，自然出现在这种情况下。出于这些结果，我们回顾了概括的复杂的树的$k$ -树的Hanlon，我们提出了另一个，在嵌套集合复杂的背景下，更自然，概括。
A LL-lattice reformulation of arithmetree over planar rooted trees. Part II
我们继续我们的自由树状代数的重新制定，处理这一次的自由树状三代数生成Y平面有根树。我们提出了一个“变形”的矢量编码中使用的第一部分，给出了一个LL格根平面树根据A。Blass和B. E.萨根树上的三个主要操作变得明确，从而为P. palacios和M.朗科我们的括号框架允许一个更容易处理的重新制定，以探索底层格描述操作的属性，并简化证明一个基本定理有关的算术树，所谓的arithmetree。算术树则被视为（N，+，x）的非交换扩展，整数由花冠扮演。我们还给出了超Catalan数或Schroder数的两种表示。
A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees
本文研究了基因树与物种树关系下祖先配置的格结构，通过迭代图的笛卡尔积定义祖先配置的有向图，并利用其与标记历史的双射关系，推导出各类树家族中标记历史的数量公式，进一步证明该格结构可扩展到非匹配树对，并用于确定固定G下的最大祖先配置对。
Spanning-Tree Maximization Forces Isoperimetry in the Square Lattice
通过拉普拉斯乘积公式和双曲坐标，证明在固定顶点数的矩形网格图中，边长越平衡生成树数量越多，其中n×n正方形网格生成树最多。
On the entropy of spanning trees on a large triangular lattice
用两种不同的方法，一种是代数的，一种是图形的，计算了一个大的三角形格上的生成树的熵S_{tri}的二重积分。两种方法都得到了相同的结果S_{tri} = [1/（2 Pi）]^2 \int_0^{2 Pi} dθ \int_0^{2 Pi} d θ ln [6-2 cos（θ）-2 cos（θ+ π）] = [3（\sqrt 3）/Pi]（1 - 5^{-2} + 7^{-2} - 11^{-2} + 13^{-2} -.）
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock和Wu给出了欧几里得格中生成树数量的指数增长率的数值。我们给出了一种新的技术，数值计算，提供更精确的值，连同严格的界限的准确性。特别是，新的价值观解决了他们的一个问题。
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for $i=0$) in bijection to various objects, like even trees, ternary trees, etc.
The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains
针对二分叉谱系无法准确反映部分样本祖先历史的问题，研究多叉有秩树形空间的递归计数、格结构及相关马尔可夫链的混合时间界，并开展模拟对比。
We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.
Middle orders: all distributive lattices between weak and Bruhat
本文研究Coxeter群中位于弱序与Bruhat序之间的分配格（称为中间序），在A型中通过二叉树构造了所有这样的格，并推广到其他Weyl群得到“极小中间序”。
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are new, some have appeared before. The lattice path models we treated, are: Hoppy's walks, the combinatorics of sequence A002212 in \cite{OEIS} (skew Dyck paths, Schröder paths, Hex-trees, decorated ordered trees, multi-edge trees, etc.) Weighted unary-binary trees also occur, and we could improve on our old paper on Horton-Strahler numbers \cite{FlPr86}, by using…
There is a long tradition of the axiomatic study of consensus methods in phylogenetics that satisfy certain desirable properties. One recently-introduced property is associative stability, which is desirable because it confers a computational advantage, in that the consensus method only needs to be computed "pairwise". In this paper, we introduce a phylogenetic consensus method that satisfies this property, in addition to being "regular". The method is based on the introduction of a partial order on the set of rooted phylogenetic trees, itself based on the notion of a hierarchy-preserving map …
An example of generalized Schur operators involving planar binary trees
杨氏格是微分偏序集的一个典型例子。微分偏序集具有罗宾逊对应，即置换和具有相同形状的标准tableaux对之间的对应，如杨氏格的情况。Fomin引入了广义Schur算子，将微分偏序集上的Robinson对应方法推广到Robinson-Schensted-Knuth对应，即某些矩阵与具有相同形状的半标准表对之间的对应。本文在以平面二叉树集为基的向量空间上引入算子。为了证明这些算子是广义Schur算子，我们构造了一个对应，它是Fomin的r-对应的推广.
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These obje…
Zeta Functions Of Discrete Groups Acting On Trees
本文推广了Bass关于均匀树格zeta函数的工作。利用冯·诺依曼代数理论，建立了一种机器来定义有界度树的离散自同构自同构群的zeta函数。主要定理涉及zeta函数的决定因素的运营商定义的边缘或顶点的树。定义了一个与具有适当Hilbert表示的非均匀树格相关联的zeta函数。定义了具有有限余体积群作用或余紧群作用的无限图的Zeta函数。
We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $Γ$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{Γ,T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{Γ,T}$ modules is isomorphic to the Euclidean lattice $\mathbb Z^{E(Γ)}$, and we describe the sublattices of integer cuts and integer flows on $Γ$ in terms of the representation theory of $A_{Γ,T}$.…
Nested set complexes of Dowling lattices and complexes of Dowling trees
给定一个有限群G和一个自然数n，我们研究了相应的Dowling格Q（G）的套集复形及其最近由Hultman引入的G-对称分拆的子集Q_G的复形的结构，以及G-对称系统树复形T_G的结构. Hultman证明了T_G和Q_G是同伦等价的和Cohen-Macaulay的，并确定了它们的顶同调的秩.应用Feichtner和Kozlov的建集和套集复形理论，证明了T_G实际上是被Q_G的序复形再分的.引入了Dowling树复形T（G），证明了它被Q（G）的阶复形细分，并包含T_G作为子复形.我们证明了T（G）是由T_G通过在某些子复形上连续锥化而得到的。我们显式地独立地计算了从T_G到T（G）的过程中增加了多少个同调球。
Merge Trees of Length-Filtered Lattice Knot Spaces
本文引入格滤子移动图作为纽结类型的有限状态实验模型，通过BFACF移动系统研究简单立方晶格中8字结的离散合并尺度，并给出合并路径。
In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a combinatorial model for torsion pairs using polyogonal subdivisions of a convex polygon. We use this model and the lattice theory to classify 2-term simple-minded collections in the bounded derived category of the corresponding tiling algebra. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver. Our m…
Exponents and bounds for uniform spanning trees in d dimensions
均匀生成树是一种统计模型，它是通过在给定的图（例如d维立方晶格的一部分）上取所有生成树的集合而获得的，每个不同的树具有相等的概率。这类树的一些性质可以通过图上的拉普拉斯矩阵，利用格拉斯曼积分得到。我们用它来获得精确的指数，这些指数的幂律衰减的概率，树的k个不同的分支通过接近每两个不同的点，作为网格的大小趋于无穷大。
We show that the subgraph induced in Young's graph by the set of partitions with an odd number of standard Young tableaux is a binary tree. This tree exhibits self-similarities at all scales, and has a simple recursive description.
We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. By relying on the interplay between bipartite distance hereditary graphs and series-parallel graphs, we show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal tim…
Among subgraphs with a fixed number of vertices of the regular square lattice, we prove inequalities that essentially say that those with smaller boundaries have larger numbers of spanning trees and vice-versa. As an application, we relate two commonly used measurements of the compactness of district maps.
A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on $n$ leaves, sample sequences of length $Ω(\log n)$ are needed. It was conjectured by M. Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than $p^{\ast} = (\sqrt{2}-1)/2^{3/2}$, then the tree can be recovered from seque…
Random Planar Lattices and Integrated SuperBrownian Excursion
在本文中，一个令人惊讶的连接之间的一个特定品牌的随机格，即平面四边形，和奥尔德斯的集成超布朗漂移（ISE）。结果表明，n个面的随机四边形的半径r_n收敛于一维ISE的支集的宽度r=R-L.更一般地，在随机四边形中到随机顶点的距离的分布在其缩放极限中由随机测量ISE移位以将其支持的最小值设置为零来描述。第一个组合成分是嵌入在正半线的树的四边形的编码，让人想起科里和Vauquelin的标记良好的树。第二步涉及这些树嵌入（离散）树的意义上的奥尔德斯，通过共轭树的原则，一个类似的树木的韦尔瓦特的建设布朗漂移的桥梁。从概率论，我们需要一个新的结果独立的利益：随机嵌入平面树的编码的两个轮廓行走的ISE的布朗蛇描述的弱收敛。我们的结果表明，存在一个连续随机映射，它用ISE描述了二维纯量子引力中动力学三角剖分的标度极限。
Critical scaling profile for trees and connected subgraphs on the complete graph
通过分析完全图上树和连通子图的生成函数，在临界窗口内识别出两者共有的单一标度轮廓，并推测该轮廓适用于高维离散环面（d≥8）上的格点树和格点动物。
Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the normalized volumes of the UPGMA cones for small n. Keywords: phylogenetic trees, polyhedral combinatorics, partition lat…
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