The length of the longest common subsequence of two independent mallows permutations

Jin, Ke

doi:10.1214/17-aap1351

Cited by 7 publications

(12 citation statements)

References 29 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Aspects of the Mallows distribution that have been studied include the longest increasing subsequence [3,5,27], longest common subsequences [23], pattern avoidance [9,10,28], the number of descents [19] and the cycle structure [14].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Cycles in Mallows random permutations

Müller¹,

Verstraaten²

2022

Preprint

View full text Add to dashboard Cite

We study cycle counts in permutations of 1, . . . , n drawn at random according to the Mallows distribution. Under this distribution, each permutation π ∈ S n is selected with probability proportional to q inv(π) , where q > 0 is a parameter and inv(π) denotes the number of inversions of π. For fixed, we study the vector (C 1 (Π n ), . . . , C (Π n )) where C i (π) denotes the number of cycles of length i in π and Π n is sampled according to the Mallows distribution. When q = 1 the Mallows distribution simply samples a permutation of 1, . . . , n uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means 1, 1 2 , 1 3 , . . . , 1 . Here we show that if 0 < q < 1 is fixed and n → ∞ then there are positive constants m i such that each C i (Π n ) has mean (1 + o(1)) • m i • n and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when q > 1 there is a striking difference between the behaviour of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behaviour depends on the parity of n when q > 1. Both (C 1 (Π 2n ), C 3 (Π 2n ), . . . ) and (C 1 (Π 2n+1 ), C 3 (Π 2n+1 ), . . . ) have discrete limiting distributions -they do not need to be renormalized -but the two limiting distributions are distinct for all q > 1. We describe these limiting distributions in terms of Gnedin and Olshanski's bi-infinite extension of the Mallows model.We investigate these limiting distributions further, and study the behaviour of the constants involved in the Gaussian limit laws. We for example show that as q ↓ 1 the expected number of 1-cycles tends to 1/2 -which, curiously, differs from the value corresponding to q = 1. In addition we exhibit an interesting "oscillating" behaviour in the limiting probability measures for q > 1 and n odd versus n even.

show abstract

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Cycles in Mallows random permutations

Müller¹,

Verstraaten²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 3 is used in our proof (see [7]) of a weak law of large numbers for the length of the longest common subsequence of two independent Mallows permutations. Theorem 3 is also of interest in its own right because it provides us a nontrivial example for a more general question: if we have two independent sequences of random permutations {π n } and {τ n } whose limiting empirical measures are known, under what condition does the limiting empirical measure of {τ n • π n } exist with density of a similar form as ρ in (4)?…”

Section: Resultsmentioning

confidence: 99%

“…The grid representation of τ can be obtained by removing the third row from the grid of π and reinserting it between the sixth row and seventh row of the grid of π (see Figure 1). π = (4, 1, 7, 3, 6, 2, 5) (3,1,7,6,5,2,4) From this definition, it can be seen that |Q(π, i)| = n for any π ∈ S n . Also, for any π, τ ∈ S n , we have either Q(π, i) = Q(τ, i) or Q(π, i) ∩ Q(τ, i) = ∅.…”

Section: Definitionmentioning

confidence: 99%

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Jin

2019

J Theor Probab

Self Cite

View full text Add to dashboard Cite

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 show the convergence of the random empirical measure of the product of two independent permutations drawn from the Mallows measure, when q is a function of n and n(1 − q) has limit in R as n → ∞.Keywords Mallows measure · random permutation · convergence of measure Mathematics Subject Classification (2010) 60F05 · 60B15 · 05A05 1 − q i 1 − q .

show abstract

“…The longest subsequence common to all sequences in a set of sequences is called a longest common subsequence (LCSS) [15]. The most efficient way to find the LCSS is dynamic programming, which first recursively computes the length of common subsequences and then identifies the common subsequence with the greatest length.…”

Section: Construction Of Social Network Based On Trajectory Datamentioning

confidence: 99%

Prediction of English Scores of College Students Based on Multi-source Data Fusion and Social Behavior Analysis

Zhao¹,

Ren²,

Zheng³

2020

RIA

View full text Add to dashboard Cite

Multi-source data fusion is the premise of applying big data technology in specific fields. Inspired by the theory on multi-source data fusion, this paper fuses various data on college students, including motion trajectories, consumptions, and social behaviors, and adopts support vector machine (SVM), a machine learning (ML) classifier to predict the English scores of college students. The behavior trajectories were taken into account, because this type of data represents the social similarity between students. Specifically, the behavioral features of college students were extracted, and subject to principal component analysis (PCA). Based on these features, the correlation between student score and social relation was analyzed, and used to predict the English scores of college students. Experimental results show that our method can accurately reflect the relationship between the social behaviors and course scores of college students.

show abstract

The length of the longest common subsequence of two independent mallows permutations

Cited by 7 publications

References 29 publications

Cycles in Mallows random permutations

Cycles in Mallows random permutations

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Prediction of English Scores of College Students Based on Multi-source Data Fusion and Social Behavior Analysis

Contact Info

Product

Resources

About