Asymptotic Enumeration of Labelled Interval Orders
Graham Brightwell, Mitchel T. Keller
在Zagier、Bousquet-Mélou等人工作的基础上，和Khamis的结果，给出了$n$ -元集上标号区间序个数的一个渐近公式.
Building on work by Zagier, Bousquet-Mélou et al., and Khamis, we give an asymptotic formula for the number of labelled interval orders on an $n$-element set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On the enumeration of labelled hypertrees and of labelled bipartite trees
我们给出了一个简单的公式的数量的超树与$k$超边的大小和$n+1$标记的顶点与规定的程度。这个公式的一个轻微的推广计数标记的二分树与规定的程度在每一类顶点。
We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Turán graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
Asymptotic enumeration of labelled graphs with a given genus
我们列举了根2-连通和3-连通的表面映射的顶点和边缘。我们还推导出随机3连通映射的大面宽结果的二元版本。然后利用这些结果导出亏格g（1-连通、2-连通和3-连通图）标号图个数的渐近公式。
Labelled and unlabelled enumeration of $k$-gonal 2-trees
本文通过用四边形、五边形或$k$边多边形代替三角形（$k$-边）来推广2-树，其中$k\geq 3$是固定的.这种推广，$k$-树2，是自然的，是密切相关的，在平面的情况下，一些专门的细胞生长问题。我们的目标是根据$k$ -树的个数$n$，对$k$ -树2-树进行标号和非标号的枚举.我们给出明确的公式在标记的情况下，并在未标记的情况下，递归和渐近公式。
This work is a follow-up of the article [Proc.\ London Math.\ Soc.\ 119(2):358--378, 2019], where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$ vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} ρ^{-n} n!$, where $g>0$ is a constant and $ρ\approx 0.24377$ is the radius of convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of several classes of…
We give explicit formulas enumerating 4-regular labelled and unlabelled one-face maps.
Combinatorics of Labelled Parallelogram polyominoes
我们得到了标记的多角形的计数的显式公式。这些是由从原点到点（k，n）的东北格子路径在上下两个方向上界定的四角形。从1到n的数字（标签）被双射地附加到上边界路径的$n$北台阶，条件是它们沿着连续的北台阶以递增的值出现。我们计算了对称群S_n在这些标号上作用的Frobenius特征。所有这些枚举结果进行了改进，以考虑到这些polyominoes的面积。我们把我们的计数结果与积分Macdonald多项式是联合本征函数的算子理论联系起来。我们还解释了如何这些相同的多项式可以用来显式地构造一个SL_2-不变量环的线性基。
Enumeration of Permutation Classes and Weighted Labelled Independent Sets
本文通过阶梯编码将排列类映射为加权独立集，推导生成函数，建立双射以枚举排列类，并揭示不平衡Wilf等价性及随机采样方法。
Enumeration and limit laws of series-parallel graphs
证明了$n$顶点上的标号串-平行图的个数$g_n$是渐近的$g_n \sim g\cdot n^{-5/2} γ^n n!$，其中$γ$和$g$是显式的可计算常数.我们证明了随机串-平行图的边数是渐近正态的，其均值和方差都是线性的，并且边数集中在它的期望值附近。类似的结果证明了标记的外平面图和不包含$K_{2,3}$作为一个小的图。
We consider the counting problem of the number of \textit{leaf-labeled increasing trees}, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained under certain population-genetical models such as multiple-merger coalescents. While the combinatorics of the binary trees among those are well understood, for the number of all trees only an approximate asymptotic formula is known. In this work, we validate this formula up to constant terms and compare the asymptotic behavior of the number of all leaf-labell…
Asymptotic enumeration and limit laws of planar graphs
给出了$n$顶点上标号可平面图个数的一个渐近估计。我们还发现了随机可平面图中边数、连通分量数和其他参数的极限律。
The structure and labelled enumeration of K_{3,3}-subdivision-free projective-planar graphs
本文研究了一类2-连通非平面K_{3,3} -无子剖分图，它们可嵌入到射影平面中.我们证明了这些图允许唯一的分解为图K_5（核心），其中的边被由2-连通可平面图构造的两极网络所取代。本文描述了在标号情况下枚举这些图的一种方法。此外，我们还列举了F中的同胚不可约图和同胚不可约2-连通可平面图。特别使用的是两极定向串并联网络。我们还证明了当n ≥ 6时，F中n阶图的边数m满足m <= 3 n-6的界.
We present upper and lower bounds for the number $i_n$ of interval graphs on $n$ vertices. Answering a question posed by Hanlon, we show that the ordinary generating function $I(x) = \sum_{n\ge 0} i_n\,x^n$ for the number $i_n$ of $n$-vertex interval graphs has radius of convergence zero. We also show that the exponential generating function $J(x) = \sum_{n\ge 0} i_n\,x^n/n!$ has radius of convergence at least $1/2$.
A new counting methods, including the issue of counting labelled self-complementary graphs
Harary和Palmer在他们的书（Graphical Enumeration，Academic Press，New York and London，1973）的结尾宣布了标记自补图的计数问题。本文解决了这一问题。解决这个问题的一个方法导致下列公式的推导：（a）具有给定阶的自同构群的标号图的数目的公式。(b)关于具有给定自同构群阶的未标号图的个数的公式。(c)给定自同构群阶的标号自补图的个数的一个公式。(d)具有给定阶自同构群的无标号自补图的个数的一个公式。
Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Préville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection …
Enumeration of interval graphs and $d$-representable complexes
本文研究了$d$-可表示单纯复形数量的渐近估计，涵盖了$d=1$时的区间图计数问题，并得出$d$-可表示复形在$d$-可坍缩复形中占比极小的结论。
Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks
本文通过构造双射分别将两条链之积偏序集的区间闭集映射到双色Motzkin路径，将截断矩形偏序集的区间闭集映射到四分仪游走，从而求得了对应的生成函数与函数方程。
Asymptotic enumeration of sparse 2-connected graphs
本文给出了$n$顶点和$m$边的标号2-连通（简单）图的个数的一个渐近公式，条件是$m-n\to\infty$和$m=O(n\log n)$为$n\to\infty$ .这是先前结果未涵盖的整个$m$范围。证明涉及确定最小度至少为2的随机图的核和核的性质。2-边连通性的情况也被类似地处理。对于大多数（“典型”）序列，我们还得到了具有给定度序列的2-连通图的个数的计算公式.我们的主要结果解决了Wright在1983年提出的一个问题。
The enumeration of simple permutations
一个简单的置换是一个没有适当的非单点区间映射到一个区间。本文从几个方面考虑了简单置换的计数问题。我们的结果包括一个简单的排列和普通的生成函数之间的直接关系，这一系列的系数是不是P-递归，这些系数的渐近展开，和一些同余结果。
Counting connected multitype graphs with prescribed edge counts
研究稀疏标记连通多类型图的渐近计数，通过非齐次随机图的大偏差原理和连通性概率，将计数问题转化为巨分支问题，推广了Bender、Canfield和McKay的经典结果。
Following a result of Eriksen and Sjöstrand (2014) we detail a technique to construct structures following the Fishburn distribution from appropriate Mahonian structures. This technique is introduced on a bivincular pattern of Bousquet-Mélou et al. (2010) and then used to introduce a previously unconsidered class of matchings; explicitly, zero alignment matchings according to the number of arcs which are both right-crossed and left nesting. We then define a statistic on the factorial posets of Claesson and Linusson (2011) counting the number of features which we refer to as mislabelings and de…
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …
Oriented Interval Greedoids
本文提出了一个“有向区间拟阵”的定义，它同时推广了有向拟阵的概念和L. J. Billera，S. K. Hsiao和J.S. Provan在“凸几何和相关的多面体细分领域的枚举”[离散计算。39（2008），no. 1 - 3，123 - 137]。对于有向拟阵，每个有向区间拟阵都对应一个球面单纯复形，其面的计数仅依赖于其下的区间拟阵。
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
我们描述了2-连通非平面环形图的结构，没有K_{3,3} -细分，使用适当的替代平面网络到某些图形的边缘称为环形核心。结构结果是基于[A. Gagarin和W. Kocay，“包含K_5-子划分的嵌入图”，Ars Combin. 64（2002），33-49]。它允许在线性时间内识别这些图，并且可以通过使用类似于[A]的方法来枚举不包含K_{3,3} -细分并且具有最小顶点度2或3的标记2-连通环形图。Gagarin，G. Labelle和P. Leroux，“Counting labeled可平面图without a K_{3,3} -subdivision”，提交，arXiv：math.CO/0406140，（2004）]。
Asymptotic enumeration and limit laws for graphs of fixed genus
证明了可嵌入亏格为g的可定向曲面S_g中的n阶标号图的个数以类似于$c^{(g)}n^{5(g-1)/2-1}γ^n n!$的方式渐近增长，其中$c^{(g)}>0$，$γ\approx 27.23$是可平面图的指数增长性.这推广了由Gimenez和Noy得到的平面g=0的结果。对非可定向曲面也得到了类似的结果。此外，证明了几个感兴趣的参数的行为渐近的平面的情况下。特别地，可嵌入到S_g中的随机图具有唯一的线性大小的2-连通分支的概率很高。
Enumeration of flattened $k$-Stirling permutations with respect to descents
本文证明扁平化k-Stirling排列与类型B集合划分的着色变体双射对应，并用加权标记结构给出其基数和下降多项式的指数生成函数及若干计数公式。
Asymptotic enumeration of 2-covers and line graphs
在本文中，我们发现$n$标记的顶点和不同类型的相关组合对象称为2-覆盖的线图的数量的渐近枚举。我们发现$[n]$上的2-覆盖数$s_n$和真2-覆盖数$t_n$都具有渐近增长性$$ s_n\sim t_n\sim B_{2n}2^{-n}\exp(-\frac12\log(2n/\log n))= B_{2n}2^{-n}\sqrt{\frac{\log n}{2n}}, $$，其中$B_{2n}$是$2n$的第n个Bell数，而$[n]$，$v_n$和线图$l_n$上的限制2-覆盖数$u_n$，限制真2-覆盖数$v_n$和真2-覆盖数$t_n$都具有渐近增长性.都有增长$$ u_n\sim v_n\sim l_n\sim B_{2n}2^{-n}n^{-1/2}\exp(-[\frac12\log(2n/\log n)]^2). $$在我们的证明中，我们使用概率参数的非限制类型的2-覆盖和生成函数的方法的限制类型的2-覆盖和线图。
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$…
Enumeration of maps with the Dumitriu-Edelman model
利用Dumitriu-Edelman三对角模型，给出了β系综粒子幂和累积量在1/N和β下的展开，系数由Bouttier、Fusy和Guitter引入的适当标记地图表示，仅涉及可定向地图，并通过新映射将前两阶与LaCroix的展开联系起来。
We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold graphs, we also count by number of dominating vertices, and for loop-threshold graphs we count by number of looped dominating vertices. We also obtain an analog of the Frobenius formula (connecting Eulerian numbers and Stirling numbers of the second kind) in the context of labelled threshold graphs.
Efficient enumeration of non-isomorphic interval graphs
本文改进了非同构区间图的枚举算法，将时间延迟从 $O(n^4)$ 降至 $O(n^3 \log n)$，并扩展了目录至 $n$ 不超过 15 的所有图。
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree familie…
In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2…
A multivariate hook formula for labelled trees
最近文献中出现了几个二叉树的钩和公式。本文给出了一个类似的公式，无序增加树的大小为r，其中涉及r参数。右边可以很好地写为线性因子的乘积。我们研究了这个新公式的两个特殊化，包括Cayley关于顶点度的树的枚举。给出了Hook公式的三个证明。这些证据之一产生有点间接，从表示论的对称群，特别是使用Kerov的字符多项式。其他的证据是更直接的，和独立的利益。
Tamari lattices and parking functions: proof of a conjecture of F. Bergeron
大小为n的m-ballot路径是正方形网格上的路径，由北向和东向单位步长组成，从（0，0）开始，结束于（mn，n），并且永远不会低于线{x=my}。这些路的集合可以具有一个格结构，称为m-Tamari格，记为T_n^（m），它推广了通常的Tamari格T_n（m=1）。这种格是由F.伯杰龙与共不变空间的研究。他提出了几个有趣的公式处理枚举的间隔在这个格子。其中之一是T_n^（m）中的区间数为$$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$。这个猜想最近得到了证明，但不是以双射的方式，而它的形式强烈地暗示了与平面树的联系。在这里，我们证明了另一个猜想的Bergeron，其中涉及的数量标记，间隔。T_n^（m）的区间[P，Q]被标记，如果Q的北阶被标记为从1到n，使得标记沿着任何连续的北阶序列增加。本文证明了T_n^（m）中标号区间的个数为$$ {(m+1)^n(mn+1)^{n-2}}. $$。这些数的形式表明了与停车函数的联系，但我们的证明是非双射的。它是基于一个递归的描述的间隔，这转化为一个函数方程满足相关的生成函数。这个方程涉及一个导数和一个均差除法，是关于两个附加变量的。解这个方程是这篇论文最难的部分。找到一个双射证明仍然是一个悬而未决的问题。
The representation of the symmetric group on m-Tamari intervals
本文研究对称群S_n在m-Tamari格带标记区间上的自然表示，证明其特征标的显式公式，并得到带标记m-Tamari区间数为(m+1)^n(mn+1)^{n-2}，通过求解涉及催化变量的生成函数方程完成证明。
Asymptotic enumeration of integer matrices with constant row and column sums
设s，t，m，n是正整数，使得sm=tn。设M（m，s;n，t）是{0，1，2，.}上的m × n矩阵的个数。其中每一行求和为s，每一列求和为t。当量，M（m，s; n，t）对m × n阶的2路列联表进行计数，使得行边缘和都是s，列边缘和都是t。第三个等价的描述是M（m，s;n，t）是具有m个s度顶点和n个t度顶点的半正则标号二部多重图的个数。当m=n和s=t时，这样的矩阵也被称为n × n幻方，其线和等于t。我们证明了一个精确的渐近公式M（m，s;n，t），这是有效的范围内（m，s;n，t），其中m，n成为无穷大，而保持近似相等，平均条目不是太小。该范围包括m/n、n/m、s/n和t/m从下面限定的情况。
On naturally labelled posets and permutations avoiding 12-34
本文建立了{3, 2+2}-free自然标号偏序集与避免12-34排列间的双射，通过级数分析探究其渐近增长，并揭示了3-free自然标号偏序集与其他组合对象及区间序层级结构的联系。
We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders…
Counting Self-Dual Interval Orders
本文提出了一种新的方法来推导区间阶的生成函数的公式，这些生成函数是按其大小、大小和最小、极大元的个数来计算的。我们的方法使我们不仅可以推广以前的结果一般区间订单的精细枚举，但也枚举自对偶区间订单类似的统计。利用新导出的生成函数公式，我们能够证明自对偶区间阶与无零行的上三角形矩阵之间的双射关系。以前，在一般区间序和没有零行和零列的上三角形矩阵之间已经建立了类似的双射关系。
Interval structure of the Pieri formula for Grothendieck polynomials
我们给出了一个组合的解释Pieri公式的双重Grothendieck多项式的Bruhat阶的间隔。另一个描述已给予Lenart和Postnikov在链枚举方面。我们使用Lascoux的解释产品的Grothendieck多项式的产品的两种生成器的0-Hecke代数，或排序操作。用这种方法我们得到了Lenart和Postnikov结果的一个直接证明，并证明了该结果中出现的置换集实际上是一个Bruhat序区间.
A random binary search tree grown from the uniformly random permutation of $[n]$ is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction $c_k$ of vertices of a fixed rank $k\ge 0$ is shown to decay exponentially with $k$. Notoriously hard to compute, the exact fractions $c_k$ had been determined for $k\le 3$ only. We computed $c_4$ and $c_5$ as well; both are ratios of enormous integers, denominator of $c_5$ being $274$ digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor…
Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in ea…
We present a unified general method for the asymptotic study of graphs from the so-called "subcritical"$ $ graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class ${G}=\cup_n {G_n}$ satisfies asymptotically the…
Asymptotic enumeration of minimally transitive permutation groups
该研究证明了极小传递置换群计数的Pyber上界在固定素数幂序列上最优，推导得出顶点传递图和有向图的数量阶，并给出Pyber上界的未发表证明，探讨对McKay–Praeger猜想的启示。
Interval and $\ell$-interval Rational Parking Functions
本文推广停车函数至区间有理停车函数，并引入 $\ell$-区间限制，给出计数公式、递归枚举及与 Fubini 排名和杠偏好排列的联系。
We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetr…
Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps
通过建立贪婪Tamari区间与二部平面地图的同构递归分解，给出了贪婪1-Tamari区间与平面2-星座等计数的组合证明，并提出了更一般的猜想。
Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\mathfrak{S}_n$
本文通过单位Fubini排名与停车函数的联系，完整刻画并枚举了对称群弱序中的布尔区间总数及秩为k的区间数，并证明以任意元素为最小元素的布尔区间数为斐波那契数乘积。
For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects paired with their multiplicities. For example, $\mathrm{MSET}(\mathbb{N})$ is (isomorphic to) the class of number partitions of positive integers, a prominent and well-studied case. The multiset construction appears naturally in the study of unlabelled objects, for example graphs or various structures related to number partitions. Our main result establishes…
Enumeration of super-strong Wilf equivalence classes of permutations in the generalized factor order
本文通过递归公式枚举广义因子序下排列的超强Wilf等价类，并揭示其与移位等价的联系，从而完整描述并枚举排列的移位等价类。
The cd-index of Bruhat intervals
通过结构递归研究Coxeter群Bruhat序区间中的旗帜计数，推导出Bruhat区间是PL球体的新证明及cd指数递推公式，并提出Bruhat球体比多面体更小的猜想。
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic …
Boolean intervals in the weak Bruhat order of a finite Coxeter group
本文对所有有限Coxeter群统一刻画了弱Bruhat序中的布尔区间，给出计数公式，并推广到C_n和D_n型，恢复并扩展了A_n型的已知结果。
Explicit enumeration and large-valence asymptotics of even-valent maps
本文推导了具有固定最小嵌入亏格的规则偶价图数量的显式公式，允许价参数和顶点数变化，扩展了亏格0和1的结果，并给出了亏格2至4的显式计数及大价渐近行为。
The Stanley-Stembridge Conjecture for $\bf 2 + 1 +1$-avoiding unit interval orders: a diagrammatic proof
本文用图解方法重新证明了 Stanley-Stembridge 猜想在避免模式 $\f 2+1+1$ 的单位区间序上的特例，该特例最初由 Gebhard 和 Sagan 证明。
Exact and asymptotic enumeration of unrestricted binary phylogenetic networks through automorphism weights
本文通过自同构权重对无限制二元系统发育网络进行精确与渐近计数，提出递归方法，给出k≤5的闭式计数，并证明非树-子网络占比及随机网络非平凡自同构概率的渐近结果。
Paper BOAT
本文提出了一种计算 affine Weyl 组下 Bruhat 区间大小的公式，并推测其可推广至所有此类组。
A circular order on edge-coloured trees and RNA m-diagrams
我们研究了一个循环顺序标记，m-边着色树与k顶点，并表明，这样的树与一个固定的循环顺序是双射与一组RNA的m-图的度k，组合对象，可以被视为RNA二级结构的某种。我们列举这些集，并表明，一组树与一个固定的循环顺序可以被描述为一个等价类的传递封闭的操作，在情况下，m=3，出现作为一个感应的上下文中的区间交换变换。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。