Andrei Khruzin
We determine the number of nonequivalent chord diagrams of order $n$ under the action of two groups, $C_{2n}$, a cyclic group of order $2n$, and $D_{2n}$, a dihedral group of order $4n$. Asymptotic formulas are also established.
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We enumerate chord diagrams without loops and without both loops and parallel chords. We show that the former ones describe Hamiltonian paths in $n$-dimensional octahedrons. The latter ones are also known as shapes. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations.
In this paper we consider the enumeration of orientable and non-orientable chord diagrams. We show that this enumeration is encoded in appropriate expectation values of the $β$-deformed Gaussian and RNA matrix models. We evaluate these expectation values by means of the $β$-deformed topological recursion, and - independently - using properties of quantum curves. We show that both these methods provide efficient and systematic algorithms for counting of chord diagrams with a given genus, number of backbones and number of chords.
A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\bf C}_g(z)=\sum_{n\geq 0} {\bf c}_g(n)z^n$ for the number ${\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\geq 1$ with a given number $n\geq 0$ of chords and find the remarkably simple formula ${\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\over 4})\neq 0$ and $R_g(0) = {\bf c}_g(2g)\neq 0.$ In particular, ${\bf C}_g(z)$ is algebraic over $\m…
We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} γ^n n!$ for a constant $c_1>0$ and $γ\approx 11.89235$. We also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} δ^n$, where $δ= 1/σ\approx 6.40375$, and $σ$ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively r…
We introduce the boundary length and point spectrum, as a joint generalization of the boundary length spectrum and boundary point spectrum in arXiv:1307.0967. We establish by cut-and-join methods that the number of partial chord diagrams filtered by the boundary length and point spectrum satisfies a recursion relation, which combined with an initial condition determines these numbers uniquely. This recursion relation is equivalent to a second order, non-linear, algebraic partial differential equation for the generating function of the numbers of partial chord diagrams filtered by the boundary …
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…
Enumeration of one-cycle signed chord diagrams and their duals for optimal Morse flows on closed surfaces
该研究将闭曲面上最优Morse流的分类构造算法化，提出枚举单圈带符号弦图及对偶的算法，经多类数据验证，可分组等价类并识别自对偶类。
We give necessary and sufficient conditions for a weight system on multiloop chord diagrams to be obtainable from a metrized Lie algebra representation, in terms of a bound on the ranks of associated connection matrices. Here a multiloop chord diagram is a graph with directed and undirected edges so that at each vertex precisely one directed edge is entering and precisely one directed edge is leaving, and each vertex is incident with precisely one undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev invariants for knots and links. The $k$-th connection matrix of a fu…
A linear chord diagram of size $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus regarding maximal chord diagrams enumerated up to rotations.
In this short note, we find the number of forests of chord diagrams with a given number of trees and a given number of chords.
We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number of short chords, where the latter is defined as all members of the matched set being adjacent, and is the generalization of a short chord or loop in a linear chord diagram. We also enumerate $k$-chord diagrams by the number of connected components built from short chords and provide the associated generating functions in this case. We show that the distribut…
We derive a functional relation between the generating functions of connected chord diagrams and 2-connected chord diagrams. This relation enables us to calculate an asymptotic expansion for the number of 2-connected chord diagrams on $n$ chords. The asymptotic information obtained from this expansion refines the last established results and provides a simple alternative for calculating the asymptotic behaviour of certain Green functions in Quenched QED and Yukawa theory in the context of quantum field theory.
Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams
本文研究线性弦图中桥的渐近分布，证明桥数渐近正态，并引入结晶图概念，给出其短弦数的渐近分布及峰值位置。
Counting Bubbles in Linear Chord Diagrams
该研究针对线性弦图，定义了无短弦的气泡区域，推导了按大小计数气泡的生成函数与平均大小的精确结果，还得到长图极限下排除无短弦图时气泡大小的渐近分布精确形式。
A Correspondence between Chord Diagrams and Families of 0-1 Young Diagrams
本文建立了弦图展开数与0-1杨图之间的对应关系，并证明n-交叉的弦展开数等于满足特定行列条件的形状为(n,n-1,…,1)的0-1杨图数量。
Statistics on Linear Chord Diagrams
本文研究线性弦图中短弦数量及特定配对数量的分布，证明短弦分布单峰且期望为1，配对分布对应二阶欧拉三角形且对数凹。
We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord…
We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest component of the random intersection graph. Denoting the number of chords by n and the number of crossings by m, when m/nlog(n) tends to a limit in (0,2/π^2), we show that the chord diagram chosen uniformly at random from all the diagrams with given parameters has a component containing almost all the crossings and a positive fraction of chords. On the othe…
We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In the present paper, we first define a new module $\mathcal{M}_2$ which is generated by chord diagrams on two circles and factored by $4$T-relations. Then we construct a "covering" map from the module of framed chord diagrams into $\mathcal{M}…
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that g_n satisfies: g_n = n/2 - Theta(ln n).
A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram $\cal C$ is defined by substituting the chords of $\cal C$ with vertices and by adding edges between two vertices whenever the corresponding two chords cross each other. Let $C_n$ and $G_n$ denote the chord diagram chosen uniformly at random from all chord diagrams with $n$ chords and the corresponding intersection graph, respectively. We analyze $C_n$ and $G_n$ as $n$ tends to infinity. In particular, we study the degree of a random vertex in $G_n$, the $k$-core of $G_n$, a…
Double-scaled SYK from boundary metrics of planar maps
本文研究一类具有特殊q变形面权重的二分平面地图模型，证明其边界度量枚举仅依赖于测地线弦图的交叉数，并与双尺度SYK模型的弦图表示一致。
Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668^n, and these can be enumerated in time O(1.8668^n). For graphs of chordality at most 5, we are able to give a bette…
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).
A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a give…
In this article we study an asymptotic expansion for $C_n$, the number of connected chord diagrams on $n$ chords. The expansion is obtained in earlier work by means of alien derivatives applied to the generating series of connected chord diagrams; we seek a combinatorial interpretation. The main outcome presented here is a new combinatorial interpretation for entry https://oeis.org/A088221 of the OEIS. We will show that https://oeis.org/A088221 counts pairs of connected chord diagrams (allowing empty diagrams). This gives a combinatorial interpretation for the part of the closed form of the as…
We study the asymptotic distributions of the number of crossings and the number of simple chords in a random chord diagram. Using size-bias coupling and Stein's method, we obtain bounds on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable, and on the total variation distance between the distribution of the number of simple chords and a Poisson random variable. As an application, we provide explicit error bounds on the number of chord diagrams containing no simple chords.
This note is dedicated to the study of a Hopf module structures on the space of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems.
Rooted connected chord diagrams form a nice class of combinatorial objects. Recently they were shown to index solutions to certain Dyson-Schwinger equations in quantum field theory. Key to this indexing role are certain special chords which are called terminal chords. Terminal chords provide a number of combinatorially interesting parameters on rooted connected chord diagrams which have not been studied previously. Understanding these parameters better has implications for quantum field theory. Specifically, we show that the distributions of the number of terminal chords and the number of adja…
We give an expression for the solution to propagator-type Dyson-Schwinger equations with one primitive at 1 loop as an expansion over rooted connected chord diagrams. Along the way we give a refinement of a classical recurrence of rooted connected chord diagrams, and a representation in terms of binary trees.
By means of color chord diagrams we establish a necessary and sufficient condition for $O$-topological equivalence of functions with one essentially critical point on oriented surfaces with edge. We also calculate the number of $O$-topologically non-equivalent functions with one essentially critical point on oriented surfaces with edge.
The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The formula that gives the right number of regions can be deduced by combinatorics reasoning using the Euler's planar graph formula, etc. My contribution in the present work is to reformulate and solve such problem in terms of a fourth order difference equation and to obtain the formula proposed by Leo Moser.
Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whet…
Series solutions for a large family of single equation Dyson-Schwinger equations are given as expansions over decorated rooted connected chord diagrams. The analytic input to the new expansions are the expansions of the regularized integrals for the primitive graphs building the Dyson-Schwinger equation. Each decorated chord diagram contributes a weighted monomial in the coefficients of the expansions of the primitives and so indexes the analytic solution in a tightly controlled way.
Methods for Analyzing RNA Pseudoknots via Chord Diagrams and Intersection Graphs
本文提出通过弦图和交集图结合距离度量$τ$的方法，用于系统分析RNA伪结结构，揭示其复杂度的量化特征。
This paper presents a construction of fibered links $(K,Σ)$ out of chord diagrams $\sL$. Let $Γ$ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of $(K,Σ)$ equals the bilinear form of the simply-laced Coxeter system $(W,S)$ associated to $Γ$; and the monodromy of $(K,Σ)$ equals minus the Coxeter element of $(W,S)$. Lehmer's problem is solved for the monodromy of these Coxeter links.
Analytic combinatorics of chord and hyperchord diagrams with $k$ crossings
本文利用解析组合学方法，研究了圆盘上具有指定交叉数的弦图等族，将交叉构型的生成函数表示为无交叉构型生成函数的有理函数，并推导了渐近结果、极限分布和随机生成器。
Let $G_n$ be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an $n$-gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of $G_n$ is asymptotic to $(n - \ln n)/2$ for $n\to\infty$. We prove a local limit theorem for the distribution of $G_n$, which implies that $G_n$ is asymptotically Gaussian, with mean $(n-\ln n)/2$ and variance $(\ln n)/4$.
We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial…
On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite
本文针对交集图为完全二部图 $K_{2,n}$ 的弦图，计算了 $\mathfrak{sl}_3$ 权系统的取值，该权系统由李代数构造并满足 4 项关系。
In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.
In this article, the enumeration of partial chord diagrams is discussed via matrix model techniques. In addition to the basic data such as the number of backbones and chords, we also consider the Euler characteristic, the backbone spectrum, the boundary point spectrum, and the boundary length spectrum. Furthermore, we consider the boundary length and point spectrum that unifies the last two types of spectra. We introduce matrix models that encode generating functions of partial chord diagrams filtered by each of these spectra. Using these matrix models, we derive partial differential equations…
Generalized chord diagrams and weight systems
本文将弦图上的权系统扩展至任意排列，提出推广Vassiliev关系的函数关系，证明$gl$-和$so$-权系统满足该关系，并分析了其性质及相关的排列Hopf代数。
We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.
Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars
本文研究闭曲面中 filoop 的亏格，通过弦图和交错图计算最小亏格，给出球和与环和的典范分解，并构造生成文法，建立稳定性性质。
We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haiman's enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of "nearly-maximally labelled" staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors…
Tubings, chord diagrams, and Dyson--Schwinger equations
本文利用有根树的二元管状结构，求解了更一般的单插入位传播子型Dyson--Schwinger方程组，获得了组合透明的级数解，并讨论了其组合联系与性质。
Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove h…
We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagra…
Polynomial mixing for polygonal side matchings
研究弦图上的马尔可夫链，通过交换弦并保持亏格不变，证明在固定亏格下该链在多项式时间内混合。
Duality for the $\mathfrak{sl}_2$ weight system
本文通过引入 shares 的双色交图、内积和加弦算子，证明了 Lando 关于 $\mathfrak{sl}_2$ 权系统在特定弦图交图上取值对偶性的猜想。
In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews' work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.
We enumerate the number of staircase diagrams over classically finite $E$-type Dynkin diagrams, extending the work of Richmond and Slofstra (Staircase Diagrams and Enumeration of smooth Schubert varieties) and completing the enumeration of staircase diagrams over finite type Dynkin diagrams. The staircase diagrams are in bijection to smooth and rationally smooth Schubert varieties over $E$-type thereby giving an enumeration of these varieties.
A method to obtain all-order asymptotic results for the coefficients of perturbative expansions in zero-dimensional quantum field is described. The focus is on the enumeration of the number of skeleton or primitive diagrams of a certain QFT and its asymptotics. The procedure heavily applies techniques from singularity analysis and is related to resurgence. To utilize singularity analysis, a representation of the zero-dimensional path integral as a generalized hyperelliptic curve is deduced. As applications the full asymptotic expansions of the number of disconnected, connected, 1PI and skeleto…
Enumeration of symmetric arc diagrams
本文针对由RNA二级结构研究引出的四类弧图，利用组合方法特别是Motzkin路与三进制字的双射，给出对称元素的计数递推关系，并分析其渐近行为以说明大规模对称结构出现概率极低。
We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions are found for an arbitrary number of external legs, and a general algorithm is implemented for this calculus. From these solutions, we calculate many asymptotics expansion terms for a simple analytical tool (Taylor expansion theorem). Our approach offers new perspectives in the realm of Feynman diagrams enumeration (or zero-dimensional quantum field theory).
Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$. This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution are expressed using two operators $D$ and $E$ subject to the relation $DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel…
Enumeration of tree-type diagrams assembled from oriented chains of edges
该研究针对离散Erdős-Rényi随机矩阵模型累积量展开中的树型图计数问题，利用改进的Prüfer码得到有向边链组装树型图的数量及加权求和的显式表达式，并推广至非规则链的情况。
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself. This result arose the question whether there is a natural way to introduce a 4-term relation on the sp…
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