GL(n,q) and Increasing Subsequences in Nonuniform Random Permutations

Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations

The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{Inv(π)}$ where $q$ is a positive parameter and $Inv(π)$ is the number of inversions in $π$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $μ_{n,q}$ and $μ_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we…

Monotone Subsequences in Locally Uniform Random Permutations

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $ρ$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any $r\ge0$, the largest union of $\lfloor r\sqrt{n}\rfloor$ de…

The length of the longest increasing subsequence of Mallows permutation models with $L^1$ and $L^2$ distances

Introduced by Mallows in statistical ranking theory, Mallows permutation model is a class of non-uniform probability measures on the symmetric group $S_n$ that depend on a distance metric $d(σ,τ)$ on $S_n$ and a scale parameter $β$. Taking the distance metric to be the $L^1$ and $L^2$ distances--which are respectively known as Spearman's footrule and Spearman's rank correlation in the statistics literature--leads to Mallows permutation models with $L^1$ and $L^2$ distances. In this paper, we study the length of the longest increasing subsequence of random permutations drawn from Mallows permut…

Permutations avoiding sets of patterns with long monotone subsequences

We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length $k-1$.

A Combinatorial Proof for 132-Avoiding Permutations with a Unique Longest Increasing Subsequence

We provide a simple injective proof that the number of 132-avoiding permutations with a unique longest increasing subsequence is at least as large as the number of 132-avoiding permutations without a unique longest increasing subsequence.

Pattern avoiding permutations and involutions with a unique longest increasing subsequence

We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.

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. …

Lengths of Monotone Subsequences in a Mallows Permutation

We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviati…

关于共轭不变的独立随机置换的最长公共子序列

On the longest common subsequence of independent random permutations invariant under conjugation

本文证明了共轭不变的独立同分布随机置换的最长公共子序列长度期望大于$\sqrt{n}$且渐近至少为$2\sqrt{n}$阶，还得出其极限波动为Tracy-Widom型的结果。

Monotone Subsequences in High-Dimensional Permutations

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $Ω_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random $k$-dimensional permutation of order $n$ is asymptotically almost surely $Θ_{k}\left(n^{\frac{k}{k+1}}\right)$.

Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $τ$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern $τ$ is monotone increasing or decreasing, or any pattern of length four.

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。

Longest alternating subsequences of permutations

The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence a,b,c,d,... is alternating if a>b<c>d<.... For instance, the expected value (mean) of as(w) for w in S_n is exactly (4n+1)/6 if n>1.

随机排列中线性大小的递增子序列与permutons的Robinson-Schensted表

Increasing subsequences of linear size in random permutations and the Robinson-Schensted tableaux of permutons

本文扩展Robinson-Schensted对应到permutons，证明其RS-表非平凡时，采样排列的递增子序列呈线性增长，并给出大偏差结果及偏微分方程。

Continuously Increasing Subsequences of Random Multiset Permutations

For a word $π$ and integer $i$, we define $L^i(π)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(π):=\max_i L^i(π)$. In this paper we estimate the expected values of $L^1(π)$ and $L(π)$ when $π$ is chosen uniformly at random from all words which use each of the first $n$ integers exactly $m$ times. We show that $\mathbb{E}[L^1(π)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as $m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that $\mathbb{E}[L(π)]$ is asymptotic to the inverse gamma function …

Longest Common Subsequences in Sets of Permutations

The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008) showed that when k>=3, every set of k permutations on [n] has a common subsequence of length at least n^{1/3}. We show that, surprisingly, this lower bound is asymptotically optimal for all constant values of k. Specifically, we show that for any k>=3 and n>=k^2 there exists a set of k permutations on [n] in which the longest common subsequence has length at m…

将排列划分为单调子序列

Partitioning permutations into monotone subsequences

本文否定了 Barber 关于排列 k-可覆盖性的猜想，证明对所有 k ≥ 3 均存在反例，且反例中所有长度至多为 (k/6)^{2.46} 的子序列都满足 k-可覆盖，而排列本身不满足。

Longest increasing subsequences in pattern-restricted permutations

Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for pattern-restricted permutations in which the pattern is any one of the six patterns of length 3. We show that the (132)-avoiding case is identical to the distribution of heights of ordered trees, and that the (321)-avoiding case has interesting connections with a well known theorem of Erd\H os-Szekeres.

Kostka Numbers and Longest Increasing Subsequences

A classical bijection relates certain Kostka numbers, the Catalan numbers, and permutations of length $n$ with longest increasing subsequence (LIS) of length at most $2.$ We generalize this bijection and find Kostka numbers which count the number of permutations of $n$ with LIS length at most $w,$ the number of permutations with $(1, \cdots, w)$ as a LIS, and other similar subsets of permutations.

On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation

We consider the distribution of the length of the longest subsequence avoiding a given pattern in a random permutation of length n. The well-studied case of a longest increasing subsequence corresponds to avoiding the pattern 21. We show that there is some constant c such that the mean value of this length is asymptotic to twice the square root of c times n and that the distribution of the length is tightly concentrated around its mean. We observe some apparent connections between c and the Stanley-Wilf limit of the class of permutations avoiding the given pattern.

Bijective enumeration of permutations starting with a longest increasing subsequence

We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two `elementary' bijective proofs of this result and of its $q$-analogue, one proof using the RSK correspondence and one only permutations.

避免长度为三的模式的随机排列中某些重复上升/下降模式的最长子序列

Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

本文研究避免模式132的随机排列中重复UUD与UUUD的最长子序列，证明其期望长度分别渐近于3n/7和4n/11，并推广至所有长度为三的避免模式。

Twins in words and long common subsequences in permutations

A large family of words must contain two words that are similar. We investigate several problems where the measure of similarity is the length of a common subsequence. We construct a family of n^{1/3} permutations on n letters, such that LCS of any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and Huynh-Ngoc. We relate the problem of constructing many permutations with small LCS to the twin word problem of Axenovich, Person and Puzynina. In particular, we show that every word of length n over a k-letter alphabet contains two disjoint equal subsequences of length cnk^{…

变异性与最长k-交替子序列长度的渐近分布

The Variance and the Asymptotic Distribution of the Length of Longest $k$-alternating Subsequences

本文通过求解随机排列中k-峰数的方差，推导出最长k-交替子序列长度的渐近方差公式，并证明了该统计量的中心极限定理。

关于单调子序列及具有宏观数量不动点的不变随机排列的RS像的一个注记

A note on monotone subsequences and the RS image of invariant random permutations with macroscopic number of fixed points

本文研究具有宏观数量不动点的不变随机排列，证明在循环总数特定条件下极限形状为VKLS形状的缩放版，并增强Tracy-Widom普适性结果。

When the law of large numbers fails for increasing subsequences of random permutations

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, \[\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.\] The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from re…

含长递增子序列的排列的递推关系

Recurrences for permutations with long increasing subsequences

该研究通过证明含长递增子序列的排列数的两个双变量递推关系，得出该序列在特定范围内为D-有限的结论，进而证明了Kauers与Koutschan2023年提出的猜想。

A probabilistic approach to the asymptotics of the length of the longest alternating subsequence

Let $LA_{n}(τ)$ be the length of the longest alternating subsequence of a uniform random permutation $τ\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(τ)$. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.

排列中的笼状子序列

Caged subsequences in permutations

本文研究排列中的最大笼状序列问题，还针对均匀随机排列、经RSK对应选取的矩形排列这两类随机排列展开相关研究。

具有大递增子序列的局部均匀随机排列

Locally uniform random permutations with large increasing subsequences

本文研究随机采样点中递增子集的最大大小，给出密度的两个充分条件，使增长速率可达到大于根号N的任意幂次，方法为网格切片分析。

Minimum number of monotone subsequences of length 4 in permutations

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all …

Applications of Symmetric Functions to Cycle and Subsequence Structure after Shuffles

Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…

通过生成树与核方法计数无长单调子序列的排列

Counting permutations with no long monotone subsequence via generating trees and the kernel method

本文通过生成树和核方法，重新证明并推广了Gessel关于无长单调子序列排列的生成函数行列式公式，并扩展到对合情形，同时跟踪固定点数。

关于所有重数均相等的词中的最长递增子序列

On longest increasing subsequences in words in which all multiplicities are equal

本文针对所有重数相等的多重集排列推广了Gessel的Bessel行列式公式，提出一个表示为Gessel型Toeplitz行列式与对称群轮换指标多项式乘积多重积分的生成函数，用于统计不含特定长度递增子序列的词。

Decreasing subsequences in permutations and Wilf equivalence for involutions

In a recent paper, Backelin, West and Xin describe a map $ϕ^*$ that recursively replaces all occurrences of the pattern $k... 21$ in a permutation $σ$ by occurrences of the pattern $(k-1)... 21 k$. The resulting permutation $ϕ^*(σ)$ contains no decreasing subsequence of length $k$. We prove that, rather unexpectedly, the map $ϕ^*$ commutes with taking the inverse of a permutation. In the BWX paper, the definition of $ϕ^*$ is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map $ϕ^*$ is the key step in provi…

排列中的子序列和

Subsequence sums in permutations

本文证明对于任意k≥3，当n足够大时，{1,2,…,n}的每个排列都包含长度为k的2-加性子序列，并给出了最小n的多项式界，同时研究了单调子序列和算术拉姆齐理论中的类似结果。

Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations

Let $P_{n,k}$ be the number of permutations $π$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $π$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\infty$-q-log-convex or $\infty$-q-log…

Longest monotone subsequences and rare regions of pattern-avoiding permutations

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permut…

On the longest k-alternating subsequence

We show that the longest k-alternating substring of a random permutation has length asymptotic to 2 (n-k) / 3.

通过RSK和evacuation获得受限子序列

Descent-restricted subsequences via RSK and evacuation

本文通过RSK和evacuation方法研究了受限子序列的统计量，揭示了其与排列记录表的关系。

Average length of the longest $k$-alternating subsequence

We prove a conjecture of Drew Armstrong on the average maximal length of $k$-alternating subsequence of permutations. The $k=1$ case is a well-known result of Richard Stanley.

Longest increasing subsequences and log concavity

Let $π$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(π)$ the length of a longest increasing subsequence of $π$. Let $\ell_{n,k}$ be the number of permutations $π$ of $[n]$ with $\ell(π)=k$. Chen conjectured that the sequence $\ell_{n,1},\ell_{n,2},\dots,\ell_{n,n}$ is log concave for every fixed positive integer $n$. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Sche…

On Permutations Avoiding Short Progressions

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can be permuted to avoid 3-term and 4-term APs. We also show that any permutation of the positive integers must contain a 3-term AP with odd common difference as a subsequence, and construct a permutation of the positive integers that does not contain any 4-term AP with odd common difference.

On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique…

过山车排列与划分数

Roller Coaster Permutations and Partition Numbers

本文研究了过山车排列的划分数，提出了一个理论上限并验证了其有效性。

关于蝴蝶排列的最长递增子序列与循环数

On the longest increasing subsequence and number of cycles of butterfly permutations

本文研究由带部分主元高斯消元法生成的随机蝴蝶排列，探讨了其最长递增子序列和循环数问题，建立了幂律界限与中心极限定理，揭示了其与典型随机排列的区别。

随机排列中最长递增子序列的渐近性

Asymptotics of the Longest Increasing Subsequence in Random Permutations

本文借助 Robinson--Schensted--Knuth 对应等经典组合工具，研究随机排列中最长递增子序列的渐近行为，证明其期望长度以 $2\sqrt{n}$ 增长，并直观统一了极限形状与波动分布的经典理论。

局部均匀随机排列极限曲面的连续性

Continuity of Limit Surfaces of Locally Uniform Random Permutations

本文研究局部均匀随机排列中递减子序列并集极限曲面的连续性，证明在密度与支集有界时变分积分存在连续极大化点，并给出二元单调函数连续化的新定理。

长递增子序列与非代数性

Long increasing subsequences and non-algebraicity

本文利用Alin Bostan的近期结果，证明了两个无限排列类序列的生成函数不具有代数性。

Hardness of Permutation Pattern Matching

Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $π$ and $τ$, represented as sequences of integers, and the task is to determine whether $τ$ contains a subsequence order-isomorphic to $π$. Bose, Buss and Lubiw proved that PPM is NP-complete on general inputs. We show that PPM is NP-complete even when $π$ has no decreasing subsequence of length 3 and $τ$ has no decreasing subsequence of length 4. This provides the first known example of PPM being hard when one or both of $π$ and $σ$ are restricted to a proper hereditary class of permutations. Thi…

利用 Schur 凹性避免块上升排列中的递增子序列

Schur-Concavity for Avoidance of Increasing Subsequences in Block-Ascending Permutations

本文研究块上升排列中避免长度为 $k+2$ 的递增模式问题，通过构造集合间双射证明其计数关于块长对称且为 Schur 凹，推广了已有等价关系。

Increasing Subsequences and Kronecker Coefficients

It has been conjectured by W. Chen that the distribution of the length of the longest increasing subsequence in a uniformly random permutation is log-concave. We propose a stronger version of this conjecture which involves the Kronecker coefficients of the symmetric group.

A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words

Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $LC_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1, \dots, n\}…

Universality for random permutations and some other groups

We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the number of occurrences of a vincular patterns satisfies a CLT for conjugation invariant random permutations with few cycles and we improve the results already known for the longest increasing subsequence. The second approach is a suggestion of a generalization to other random permutations and other sets having a similar structure than the symmetric group.