J. Cooper, J. Solymosi
Consider the following problem: how many collinear triples of points must a transversal of (Z/nZ)^2 have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n-1)/4 and (n-1)/2, and consider an analogous question for collinear quadruples. We conjecture that the upper bound is the truth and suggest several other interesting problems in this area.
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We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately connected. Via counting arguments and by studying the hypergraph of collinear triples we show a new lower bound (5q/14 + O(1)) for the number of collinear triples of a permutation of GF(q) and a new lower bound (q(q + 1)/2 + 5q/14 + O(1)) on the size of the smallest Besicovitch set in GF(q)^2. Several interesting questions about the structure of the collinear triple …
Let $α:\mathbb{Z}_n\to\mathbb{Z}_n$ be a permutation and $Ψ(α)$ be the number of collinear triples modulo $n$ in the graph of $α$. Cooper and Solymosi had given by induction the bound $\min_αΨ(α)\geq\lceil(n-1)/4\rceil$ when $n$ is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined.
Let $α:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $Ψ(α)$ be the number of collinear triples in the graph of $α$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once proved $Ψ(α)\geq(q-1)/4$ and conjectured the sharp bound should be $Ψ(α)\geq(q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true.
Permutations minimizing the number of collinear triples
本文刻画了有限域上使图中共线三元组数量最少的置换，确定字典序最小者，证实 Cooper-Solymosi 猜想，并探讨其与正交拉丁方及有限仿射平面的联系。
Collinear Interior Lattice Points in Triangles Satisfying $B(T)\in\{4,5\}$
本文基于前期$B(T)=3$的研究，结合Alder广义欧拉函数$g(k)$等，分类$B4$-、$B5$-共线整数，确定$B4$-共线整数为1、2、5，无$B5$-共线整数，揭示边界格点数量对共线性模式的影响。
Four collinear points or six visible points
研究证明任意至少880个点的有限平面点集必含四个共线点或六个两两可见点。
We investigate the question of finding a bound for the size of a $χ$-colorable finite visibility graph that has at most $\ell$ collinear points. This can be regarded as a relaxed version of the Big Line - Big Clique conjecture. We prove that any finite point set that has at least 2311 points has either 4 collinear points or a visibility graph that cannot be 5-colored.
For every $k>3$, we give a construction of planar point sets with many collinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are $n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of $n$ points in the plane that does not contain $k+1$ points on a line, but it contains at least $n^{2-\frac{c}{\sqrt{\log n}}}$ collinear $k$-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $k$-tuples in such a set, and get reasonably close to the trivial upper bound $O(n^2)$.
Let $P$ be a set of $n$ points in the plane, each point moving along a given trajectory. A {\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do not all coincide, and not all of them are collinear at all times. We show that, if the points move with constant velocity, then the number of 3-collinearities is at most $2\binom{n}{3}$, and this bound is tight. There are $n$ points having $Ω(n^3/k^4 + n^2/k^2)$ distinct $k$-collinearities. Thus, the number of $k$-collinearities among $n$ points, for consta…
On the structure of pointsets with many collinear triples
本研究在组合条件下证明含大量共线三元组的平面点集存在特殊三元组构型，证实Elekes猜想的一个情形，并借助相关技巧给出Jamison定理的密度版本。
It is proved that for $k\geq 4$, if the points of $k$-dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two consecutive points. This result is new for $4\leq k\leq 10$.
On lattice triangles satisfying $\boldsymbol{B(T)=3}$ with collinear interior lattice points
本文研究了满足 $B(T)=3$ 的格点三角形，探讨当其内部有 $k$ 个格点时，使得所有 $k$ 个边界点共线的整数 $k$ 的性质。
Large Finite Point Sets Have 4 Collinear Points or a 6-Clique
该研究证明大小不小于10^11055931的有限点集必含4个共线点或6个两两可见点，解决了大线大团猜想的首个公开情况
Convex Lattice Polygons with $k\ge3$ Interior Points
本文研究了具有 $n$ 个边界格点和 $k\geq 3$ 个共线内部格点的凸格 $n$ 边形的几何性质，提出了一种通过构造本原格三角形来添加边的方法，并给出了必要条件与次数上界，最终完整分类了存在具有 $k$ 个共线及非共线内部点的凸 $n$ 边形的正整数。
North-East Lattice Paths Avoiding $k$ Collinear Points via Satisfiability
研究确定东北格路中避免k共线点的最大点数，通过求解器枚举所有避免k共线点的路径，并改进了k=7时的路径长度。
We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].
We show that if a planar graph $G$ has a plane straight-line drawing in which a subset $S$ of its vertices are collinear, then for any set of points, $X$, in the plane with $|X|=|S|$, there is a plane straight-line drawing of $G$ in which the vertices in $S$ are mapped to the points in $X$. This solves an open problem posed by Ravsky and Verbitsky in 2008. In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings.
An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.
Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results
本研究证明足够大的平面点集必含四点共线或七点近团，并推广至色数至多五或六且含色临界边的任意图，逼近大线大团猜想。
In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many colline…
An infinite walk in $\mathbb{N}^{16}$, using only unit steps, with no three collinear points
该研究证明在16维空间中，仅用单位基向量步长可无限行走且避免三点共线。
In this ote, which has been absorbed by arXiv1702.01003, we combine a recent point-line incidence bound of Stevens and de Zeeuw with an older lemma of Bourgain, Katz and Tao to bound the number of collinear triples and quadruples in a Cartesian product in $\mathbb{F}_p^2$.
In 2014, Payne-Wood proved that every non-collinear set $P$ of $n$ points in the Euclidean plane contains a point in at least $\dfrac{n}{37}$ lines determined by $P.$ This is a remarkable answer for the conjecture, which was proposed by Erdős, that every non-collinear set $P$ of $n$ points contains a point in at least $\dfrac{n}{c_1}$ lines determined by $P$, for some constant $c_1.$ In this article, we refine the result of Payne-Wood to give that every non-collinear set $P$ of $n$ points contains a point in at least $\dfrac{n}{26}+2$ lines determined by $P$ . Moreover, we also discuss some re…
We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.
A conjecture by R. Stanley on a class of alternating permutations, which is proved by R. Chapman and L. Williams states that alternating permutations with the maximal number of fixed points is equidistributed with derangements. We extend this (type $A$) result to type $B$: We prove that various classes of alternating signed permutations with the maximal number of fixed points is equidistributed with certain types of derangements (of type $B$), respectively.
Identities for permutations with fixed points
本文通过逐次求导或积分特定矩阵的行列式，推导出具有不动点的排列所满足的恒等式。
Arithmetic functions and fixed points of powers of permutations
本文研究置换幂的不动点计数函数如何确定置换的共轭类，构造了计算共轭类的算法，并刻画了可作为不动点计数函数的算术函数。
Automatic Abelian Complexities of Parikh-Collinear Fixed Points
本文证明任意（可能擦除的）Parikh-共线态射的不动点的阿贝尔复杂度函数是自动的，并给出了生成该函数的确定性有限自动机的有效构造方法。
Counting fixed-point-free Cayley permutations
通过双种类方法得到Cayley排列的函数有向图的微分方程，推导出无不动点Cayley排列的显式公式，并证明其比例趋于1/e，与排列和自映射一致。
For permutations P and T of lengths |P|\le|T|, let t(P,T) be the probability that the restriction of T to a random |P|-point set is (order) isomorphic to P. We show that every sequence \{T_j\} of permutations such that |T_j|\to\infty and t(P,T_j)\to 1/4! for every 4-point permutation P is quasirandom (that is, t(P,T_j)\to 1/|P|! for every P). This answers a question posed by Graham.
An important theorem of Beck says that any point set in the Euclidean plane is either ``nearly general position'' or ``nearly collinear'': there is a constant C>0 such that, given n points in the plane with at most r$ of them collinear, the number of lines induced by the points is at least Cr(n-r). Recent work of Gutkin-Rams on billiards orbits requires the following elaboration of Beck's Theorem to bichromatic point sets: there is a constant C>0 such that, given n red points and n blue points in the plane with at most r of them collinear, the number of lines spanning at least one point of eac…
The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of d by n matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension d=3 and compute its homology in this case, conjecturing that this complex is shellable in gener…
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.
We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Ω\left(\min\left\{ n^{2/3}m^{2/3},\; \frac{n^{10/11}m^{4/11}}{\log^{2/11}m},\; n^2,\; m^2\right\}\right) . $$ Without the assumption on $P_2$, there exist …
The big-line-big-clique conjecture states that for all $k,\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. We show that this conjecture is false for infinite point sets, by constructing a countably infinite point set with no 4 collinear points and no 3 pairwise visible points.
In this paper we prove that among the permutations of length n with i fixed points and j excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for all given i,j<=n. We use a new technique involving diagonals of non-rational generating functions. This theorem generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof.
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
Forcing quasirandomness with 4-point permutations
该研究聚焦4点置换的拟随机强制集合规模问题，突破扰动梯度线性依赖壁垒，证明此类集合的基数至少为5，推进了置换拟随机性的组合学研究。
The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed points). Specializations of those $q$-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group.
A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied under the name of square permutations. Here, we explore permutations with a fixed number of internals points, called almost square permutations. Unlike with square permutations, a precise enumeration for the total number of almost square permutations of size $n+k$ with exactly $k$ internal points is n…
Pattern-avoiding stabilized-interval-free permutations
本文研究无稳定区间（SIF）排列对长度3经典模式及模式对的避免问题，利用不动点分布、组合论证及与有序森林的双射完成枚举，并提出相关猜想。
In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n/2-c lines determined by P, for some constant c. The following weakening was proved by Beck and Szemerédi-Trotter: every set P of n non-collinear points contains a point in at least n/c lines determined by P, for some large unspecified constant c. We prove that every set P of n non-collinear points contains a point in at least n/37 lines determined by P. We also give the best known constant for Beck's Theorem, proving that every set of n points with at most k collinear deter…
Sum rules for permutations with fixed points involving Stirling numbers of the first kind
本文提出基于第一类斯特林数的置换固定点求和规则，并推导了二项系数的求和规则，同时探讨了与贝尔数的关系。
On point and block primitive designs invariant under permutation groups
本研究提出构造有限群下不变的点本原块传递t-设计的方法，证明所有点块本原G-不变设计均可由此生成，同时指出大群下全枚举的实际可行性受计算复杂度限制。
Let $f(n,\ell)$ be the maximum integer such that every set of $n$ points in the plane with at most $\ell$ collinear contains a subset of $f(n,\ell)$ points with no three collinear. First we prove that if $\ell \leq O(\sqrt{n})$ then $f(n,\ell)\geq Ω(\sqrt{\frac{n}{\ln \ell}})$. Second we prove that if $\ell \leq O(n^{(1-ε)/2})$ then $f(n,\ell) \geq Ω(\sqrt{n\log_\ell n})$, which implies all previously known lower bounds on $f(n,\ell)$ and improves them when $\ell$ is not fixed. A more general problem is to consider subsets with at most $k$ collinear points in a point set with at most $\ell$ co…
On Cycles in Multiset Permutations, Parking Functions, and Related Structures
本文研究多重集排列和停车函数中的环，通过代数和分析方法给出精确与渐近结果，并揭示终端闭包与循环点之间的意外对应关系。
We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $π$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $σ$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Björner and Wachs.
A well-known theorem in plane geometry states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of {\em betweenness}. In this paper, we prove that in the plane with the $L_1$ (also called Manhattan) metric, a non-collinear set of $n$ points induces at least $\lceil n/2\rceil$ lines. This is an improvement of the previous lower bound of $n/37$, with substantially different proof. As a consequence, we also get the same lo…
The space T_{d,n} of n tropically collinear points in a fixed tropical projective space TP^{d-1} is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d x n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space M_{0,n}(TP^{d-1},1) of n-marked tropical lines in TP^{d-1} under the evaluation map. Thus we derive a natural simplicial fan structure for T_{d,n} using a simplicial fan structure of M_{0,n}(TP^{d-1},1) which coincides with that…
Improving bounds on packing densities of 4-point permutations
本文整合4点排列堆积密度的已知结果，改进1324和1342的下界，并为1324、1342和2413提供严格上界，所有界与真实值差距在10^-4内，主要工具为旗代数框架。
A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations
受Bsila等人关于模式避免排列中固定点和固定对猜想的启发，我们证明了涉及下降统计量的双变量精细化结果。
A permutation graph $G_π$ is a simple graph with vertices corresponding to the elements of $π$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $π$. A set of vertices $D$ is said to dominate a graph $G$ when every vertex in $G$ is either an element of $D$, or adjacent to an element of $D$. The domination number $γ(G)$ is defined as the cardinality of a minimum dominating set of $G$. A strong fixed point of a permutation $π$ of order $n$ is an element $k$ such that $π^{-1}(j)<π^{-1}(k)$ for all $j<k$, and $π^{-1}(i)>π^{-1}(k)$ for all $i>k$. In this article, we count the number …
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study fixed points of both 123- and 231-avoiding permutations. We find an exact description for a scaling limit of the empirical distribution of fixed points in term of Brownian excursion. This builds on the connections between pattern-avoiding permutations and Brownian excursion developed in Part I of this series and strengthens the recent results of Elizalde (2012) and Miner and Pak (2014) on fix…
The number of fixed points of a random permutation of 1,2,...,n has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial -- almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of 1,2,...,n, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of relate…
In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $ it contains a collinear triple. Another application studies epsilon-nets in a point-line system in the plane. We prove the existence of some geometric constructions with a new tool, the so-called Hypergraph Container Method.
Distances in Planar Integral Point Sets
研究平面非共线整点集中极小距离的分布，证明极小距离在亚指数增长下几乎一维，并给出反例说明线支撑情况不可避免。
A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. Here we generalize the concept by allowing the points not collinear with $x$ to form the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry.
Limit theorems for fixed point biased permutations avoiding a pattern of length three
本文研究了在特定偏分布下，避免三元模式排列中固定点数量的极限行为，揭示了偏参数变化时分布形态的相变特性。
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