Related papers: Lengths of Monotone Subsequences in a Mallows Perm…

The longest increasing subsequence of a random walk with mean zero and finite variance is known to be $n^{1/2 + o(1)}$. We show that this is not universal for symmetric random walks. In particular, the symmetric Ultra-fat tailed random walk…

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each,…

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…

Let $L_n$ be the length of the longest common subsequence of two independent i.i.d. sequences of Bernoulli variables of length $n$. We prove that the order of the standard deviation of $L_n$ is $\sqrt{n}$, provided the parameter of the…

In this work, we present a plethora of results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to RLIS is a sequence $S$ of $n$ real numbers and a collection $Q$ of $m$ query ranges, and for each…

We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a zero-mean, symmetric heavy tailed mass…

The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < \alpha_*(p) \leq…

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…

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th…

Permutons are probability measures on the unit square with uniform marginals that provide a natural way to describe limits of permutations. We are interested in the permuton limits for permutations sampled uniformly from certain…

We study Mallows random permutations conditioned to avoid a given pattern $\alpha$ of length~$3$. When the bias parameter is of the form $e^{\beta/n}$, we prove that these permutations converge to a non-trivial explicit deterministic…

We study the statistics of the Mallows measure on permutations in the limit pioneered by Starr (2009). Our main result is the local central limit theorem for its height function. We also re-derive versions of the law of large numbers and…

We investigate the probability for the largest segment in with total displacement $Q$ in an $N$-step random walk to have length $L$. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the…

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…

This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of…

Common intervals have been defined as a modelisation of gene clusters in genomes represented either as permutations or as sequences. Whereas optimal algorithms for finding common intervals in permutations exist even for an arbitrary number…

In this note, we studied the asymptotic behaviour of the length of the longest common substring for run-length encoded sequences. When the original sequences are generated by an $\alpha$-mixing process with exponential decay (or…

We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n. A striking feature of this expansion is that tells us that the expected value of…

Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length $n$ can be computed exactly in time $O(n\log n)$, the complexity of estimating the…

In applications such as rank aggregation, mixture models for permutations are frequently used when the population exhibits heterogeneity. In this work, we study the widely used Mallows mixture model. In the high-dimensional setting, we…