Dominique Rossin, Mathilde Bouvel
In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations.
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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.
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
We provide upper and lower bounds for the expected length $\mathbb E(L_{n,m})$ of the longest common pattern contained in $m$ random permutations of length $n$. We also address the tightness of the concentration of $L_{n,m}$ around $\mathbb E(L_{n,m})$.
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.
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.
We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case our method provides a different proof of a similar result by Tracy and Widom about longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the `colored' problem to the case of usual random permutations using certain combinatorial results and element…
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…
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.
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.
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.
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.
The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $π$ is proportional to $q^{l(π)}$ with $q > 0$ being a parameter and $l(π)$ the number of inversions in $π$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n \to \infty$.
Large deviations for the longest alternating and the longest increasing subsequence in a random permutation avoiding a pattern of length three
本文计算了避免长度为三的模式的均匀随机排列中，最长交替子序列和最长递增子序列长度的大偏差，发现交替子序列六种情况共享一个速率函数（即对称伯努利随机变量之和的速率函数），递增子序列两种情况的速率函数为其两倍。
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…
Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.
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…
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…
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。
Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear group to obtain results about the longest increasing subsequence in non- uniform random permutations.
We show that the longest k-alternating substring of a random permutation has length asymptotic to 2 (n-k) / 3.
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…
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…
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。
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…
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…
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…
Exact Sampling of Permutations with a Fixed Longest Increasing Subsequence
研究对长度为 n 且最长递增子序列长度为 k 的排列进行精确均匀采样，对于 k ∈ Θ(n) 给出期望运行时间 O(n log log n) 的直接拒绝采样器，对于任意 k 给出基于 Robinson-Schensted 对应的精确采样器，并通过行列式恒等式和 Hankel 矩矩阵优化实现。
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。
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。
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。
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.
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。
In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern $3412$ and another pattern.
We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…
Two permutations are similar if they have the same length and the same relative order. A collection of $r\ge2$ disjoint, similar subsequences of a permutation $π$ form $r$-twins in $π$. We study the longest guaranteed length of $r$-twins which are tight in the sense that either each twin alone forms a block or their union does. We address the same question with respect to a random permutation.
Given a set $Π$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(Π)$, the set of permutations of length at most $n$ avoiding the patterns in $Π$, in time $O(|S_{\le n - 1}(Π)| \cdot k + |S_{n}(Π)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $Π$ in each permutation in $S_n$. Surprisingly, when $|Π| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutat…
Given permutations $σ\in S_k$ and $π\in S_n$ with $k<n$, the \emph{pattern matching} problem is to decide whether $π$ matches $σ$ as an order-isomorphic subsequence. We give a linear-time algorithm in case both $π$ and $σ$ avoid the two size-$3$ permutations $213$ and $231$. For the special case where only $σ$ avoids $213$ and $231$, we present a $O(max(kn^2,n^2\log(\log(n)))$ time algorithm. We extend our research to bivincular patterns that avoid $213$ and $231$ and present a $O(kn^4)$ time algorithm. Finally we look at the related problem of the longest subsequence which avoids $213$ and $2…
A note on the distribution of the sum of lengths of the initial longest increasing sequences in cycles of random permutations
本文研究随机排列各循环中 ILIS 长度之和 $s_n$，在 Mansour 的期望与方差结果基础上，证明其适当归一化后随 $n\to\infty$ 弱收敛至标准正态分布。
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。
This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression …
We say that a permutation $π$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $π_a<π_b<π_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.
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。
There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and probabilistic properties of sub-permutations and to investigate the relationships between 'local' and 'global' features using the concept of pattern avoidance. First, given a pattern μ, we study how the avoidance of μ in a permutation π affects the presence of other patterns in the sub-permutations of π. More precisely, considering patterns of length 3, we solv…
On Chen's conjecture for the log-concavity of the longest increasing subsequence distribution
本文证明当$n\ge 10^{10^{65}}$时，关于最长递增子序列分布对数凹性的Chen猜想成立，证明结合了Gessel行列式公式、Tracy-Widom密度估计等方法。
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\}…
In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.
The Ulam-Hammersley problem for multiset permutations
本文研究了多重集排列中最长递增/非递减子序列的渐近行为，推广了经典的 Ulam-Hammersley 问题，并通过 poissonization 与 Hammersley-Aldous-Diaconis 粒子系统变体建立联系完成证明。
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Σ_τλ_f(τ)τ$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(π,R)$ is an occurrence of the permutation pattern $π$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(π,R)$, we have $λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$ …
We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack. We use this method to describe West-3-stack-sortable permutations, that is, permutations that are sorted by three passes through a stack. We also show how the method can be applied to the bubble-sort operator. The method requires the use of mesh patterns introduced by Brändén and Claesson (2011), as well as a new type of generalized pattern we call a decorated pattern.
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。
Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length 3 patterns on multiply restricted permutations, as well as some combinatorial interpretations for non-trivial pattern relationships.
This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.
A brief presentation of basic definitions and notation used in permutation patterns research.
We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson introduced the notion of a "generalized permutation pattern" (GP) which generalizes the concept of "classical" permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrimsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other com…
In this paper we calculate the cardinality of the set S_n(T,tau) of all permutations in S_n that avoid one pattern from S_4 and a nonempty set of patterns from S_3.
For a fixed irrational number $α$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = αi \mod 1$. Our primary result is that the boundary of the Schensted shape is approximated by a piecewise linear function with at most two slopes. This piecewise linear function is explicitly described in terms of the continued fraction expansion for $α$. Our results generalize those of Boyd and Steele, who studied longest monotone subsequences. Our proofs are based on a careful analysis of monotone sets in two-dimensional lattices.
Two-count interval representation of a permutation
本文研究排列的 $k$-计数区间表示问题，证明了排列具有 $2$-计数区间表示当且仅当其最长递减子序列长度不超过2，并将此结果应用于刻画高度为3的区间序的 $2$-计数条件。
We establish a bijection from the set of all permutations (of a given length) that avoid a pattern $q$ and a context-sensitive language.
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