Subclasses of the separable permutations

Separable d-permutations and guillotine partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between $d$-dimensional permutations and guillotine partitions of a $2^{d-1}$-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain results on a…

可分离排列上统计量的分布

Distributions of statistics on separable permutations

本文推导了可分离排列上六个经典统计量分布的函数方程，借此求出上升与下降联合分布的三次方程，并显式推导了最大值与最小值统计量任意子集的联合分布，最后提出了三个单峰性猜想。

The Moebius function of separable and decomposable permutations

We give a recursive formula for the Moebius function of an interval $[σ,π]$ in the poset of permutations ordered by pattern containment in the case where $π$ is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, ..., k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Moebius function in the case where $σ$ and $π$ are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or…

The Weak Bruhat Order and Separable Permutations

In this paper we consider the rank generating function of a separable permutation $π$ in the weak Bruhat order on the two intervals $[\text{id}, π]$ and $[π, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, π]$ and $[π, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets.

Restricted non-separable planar maps and some pattern avoiding permutations

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…

The Brownian limit of separable permutations

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".

关于可分离排列和Schröder类中的三个其他对

On separable permutations and three other pairs in the Schröder class

本文研究了由大Schröder数计数的四个模式回避排列类别的位置统计，通过结构分解和核方法推导出多变量生成函数，证明这些类别的计数由Schröder数给出，并发现与中央二项系数和Kreweras覆盖层次序列相关的三角数组。

枚举可分错位排列

Enumerating separable derangements

本研究提出多项式时间算法计算集合[n]的可分错位排列数bₙ，基于生成函数技术，证明bₙ与大Schröder数指数增长常数相同，利用前3000项推测渐近行为，部分回答Vatter的相关问题。

Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections

We investigate the connections between patterns in permutations and forbidden configurations in restricted elections, first discovered by Lackner and Lackner, in order to enhance the approach initiated by the two mentioned authors. More specifically, our achievements are essentially two. First, we define a new type of domain restriction, called enriched group-separable. Enriched group-separable elections are a subset of group-separable elections, which describe a special, still natural, situation that can arise in the context of group-separability. The exact enumeration of group-separable elec…

受平坦 POP 限制的可分排列上的统计量分布

Distribution of statistics on separable permutations restricted by a flat POP

本文研究了避免平坦偏序模式（POP）的可分排列，推导出描述最一般情形下六种经典统计量联合分布的函数方程组，并在长度为3和4的特例中分别恢复了已知结果与显式有理生成函数。

Longest Common Separable Pattern between Permutations

In this article, we study the problem of finding the longest common separable pattern between several permutations. We give a polynomial-time algorithm when the number of input permutations is fixed and show that the problem is NP-hard for an arbitrary number of input permutations even if these permutations are separable. On the other hand, we show that the NP-hard problem of finding the longest common pattern between two permutations cannot be approximated better than within a ratio of $sqrt{Opt}$ (where $Opt$ is the size of an optimal solution) when taking common patterns belonging to patter…

随机递归可分排列的置换子极限

The permuton limit of random recursive separable permutations

本文提出随机递归可分排列的马尔可夫模型，证明其几乎必然收敛于递归可分置换子，并给出该极限对象的分布刻画、模式密度公式、强度测度积分公式及其与布朗可分置换子的绝对奇异性。

短长度网格模式在可分离排列上的分布

Distributions of Mesh Patterns of Short Lengths on Separable Permutations

本文研究可分离排列中长度至多2的网格模式分布，确定长度1的所有等价类分布，并利用对称操作将长度2的等分布对划分为31个等价类，其中14个类的枚举留作开放问题。

可分离排列的类Ramsey定理

Ramsey-like theorems for separable permutations

研究无限团边着色中避免特定模式的无限制子团的存在性，证明可分离排列的避免性等价于标准模型中无限齐次集的存在，其他模式则不然。

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. …

$A$型和$B$型中可分元素的组合等价性

Combinatorial equivalence of separable elements in types $A$ and $B$

研究$A$型和$B$型可分元素的组合等价性，构造对称群与超八面体群间的双射，证明降序多项式$\gamma$-正性并推导秩生成函数显式公式。

随机可分排列中最长交替子序列的均值与方差

Mean and variance of the longest alternating subsequence in a random separable permutation

本文研究随机可分排列中最长交替子序列的长度，证明其期望渐近为(2-√2)n，并给出相应四个变体的方差渐近为(16-11√2)n/2。

The Infinite Limit of Separable Permutations

Let $P_n^{\text{sep}}$ denote the uniform probability measure on the set of separable permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $σ=\{σ_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $σ^{-1}(\mathbb{N})$\rm; that is, if $σ_i=σ_j$, $i\neq j$, then $σ_i=\infty$. Extending permutations $σ\in S_n$ by defining $σ_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. We show that $\{P_n^{\text{sep}}\…

Clustering of consecutive numbers in permutations avoiding a pattern and in separable permutations

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $σ\in S_n$ by $σ=σ_1σ_2\cdots σ_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions: $\{k,k+1,\cdots, k+l-1\}=\{σ_a,σ_{a+1},\cdots, σ_{a+l-1}\}$, for some $a$. For $τ\in S_m$, let $S_n(τ)$ denote the set of $τ$-avoiding permutations in $S_n$, and let $P_n^{\text{av}(τ)}$ denote the uniform probability measure on $S_n(τ)$. Also, let $S_n^{\text{sep}}$ denote the set of sep…

Separable elements in Weyl groups

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of Fan Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of Billey and Postnikov.

将无模式排列分解为可分离排列

Factoring Pattern-Free Permutations into Separable ones

本文证明任意避免固定模式的排列都可分解为至多常数个可分离排列的乘积，并利用排列宽度给出线性时间分解算法，进而应用于有界 twin-width 图类的一阶转导刻画。

Constructing Separable Arnold Snakes of Morse Polynomials

We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values are distinct. Thus we can associate to it an alternating permutation: the so-called Arnold snake, given by the relative positions of its critical values. We realise any separable alternating permutation as the Arnold snake of a Morse polynomial.

排列的单调网格类中的环

On cycles in monotone grid classes of permutations

本文研究行列图无多环的单调网格类排列结构，提出M-和分解，刻画良拟序条件，证明多数网格类有限基并反驳2006年猜想。

等距不变置换码与相互正交拉丁方

Isometry invariant permutation codes and mutually orthogonal Latin squares

本文利用MOLS与可分置换码的对应关系，通过对称群等距群子群及其轨道代表构造多组高距离置换码，改进n=35、48、63、96时MOLS数量下界。

B型Weyl群中的可分元素与分裂

Separable elements and splittings in Weyl groups of Type $B$

本文研究B型Weyl群中的可分元素，通过禁用模式分类可分与极小不可分带符号置换，从而证实Gaetz和Gao关于分裂结构可推广至B型Weyl群的猜想。

布朗可分离置换子（Brownian separable permutons）的最长递增子序列

The longest increasing subsequence of Brownian separable permutons

本研究证明从布朗可分离置换子采样的置换的最长递增子序列长度除以$n^{\alpha(p)}$几乎必然收敛到非平凡随机变量，其中指数$\alpha(p)$由特定方程唯一确定，并推广至布朗cographon的最大团大小。

高维排列测度：理论与应用

High-dimensional permutons: theory and applications

本文提出了$d$维排列测度的理论框架，证明了$d$维排列序列的收敛性，并确定了Schnyder wood排列和$d$-separable permutations的随机高维排列测度极限，揭示了其与随机几何对象的深刻联系。

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $γ$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.

具有双传递置换自同构群的MDS码的长度

On the lengths of MDS codes with a two-transitive permutation automorphism group

本文研究具有双传递置换自同构群的MDS码，证明当4≤k≤q−3时其长度n≤q+1，进而说明MDS猜想对这类码成立。

布朗可分离排列子序列幂律界与布朗余图同质集幂律界

Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons

本研究为布朗可分离排列的最长递增子序列和布朗余图的最大团/独立集建立幂律界，给出显式指数上下界，并通过数值模拟验证下界接近最优，方法基于碎片化过程分析。