Related papers: Lengths of Monotone Subsequences in a Mallows Perm…
Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns in…
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…
In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. For instance, we show that the…
We study the $q$-deformation of the Demazure product model from arXiv:2407.21653. Consider the longest element $w_0$ in $S_n$ written as a reduced word in simple transpositions. Independently delete each transposition with probability $1-p$…
One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix $A$ and storing the permutation matrix factor $P$ from the resulting GEPP factorization $PA=LU$. We are…
We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern $w$ of length $m$ as a subsequence in a random text of length $n$. The quantity of interest is the…
We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to…
Let $S_n$ be the set of all permutations of $\{1,2,\ldots,n\}$ and let $\sigma=(\sigma_1,\sigma_2,\ldots,\sigma_n)\in S_n$. The {\it initial longest increasing sequence} (ILIS) in $\sigma$ has length $m$ if, for $1\le m\le n-1$,…
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…
The asymptotic study of tuples of random non-increasing integers is crucial for probabilistic models coming from asymptotic representation theory and statistical physics. We study the global behavior of such tuples, introducing a new family…
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…
We study the shape of the Young diagram \lambda associated 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…
We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the…
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…
Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed…
A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for…
We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{\alpha_1, \alpha_2, \cdots, \alpha_m\}$, while the…
We investigate the probability of observing a given pattern of $n$ rises and falls in a random stationary data series. The data are modelled as a sequence of $n+1$ independent and identically distributed random numbers. This probabilistic…
We analyze the secretary problem in the case that the $n$ ranked items arrive not in uniformly random order but rather according to a certain type of Luce distribution or according to a Mallows distribution on the set $S_n$ of permutations…
Let $p$ and $q$ be probability vectors with the same entropy $h$. Denote by $B(p)$ the Bernoulli shift indexed by $\Z$ with marginal distribution $p$. Suppose that $\phi$ is a measure preserving homomorphism from $B(p)$ to $B(q)$. We prove…