Related papers: On weak twins and up-and-down sub-permutations

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…

Given a permutation w, we look at the range of how often a simple reflection s_k appears in reduced decompositions of w. We compute the minimum and give a sharp upper bound on the maximum. That bound is in terms of 321- and 3412-patterns in…

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$…

We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest…

The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$. Chung, Claesson, Dukes and Graham obtained polynomials $P_k(x)$ that can be used to determine the number of…

By an $r$-tuplet in a permutation we mean a family of $r$ pairwise disjoint subsequences with the same relative order. The length of an $r$-tuplet is defined as the length of any single subsequence in the family. Let $t^{(r)}(n)$ denote the…

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random…

A permutation array $A$ is a set of permutations on a finite set $\Omega$, say of size $n$. Given distinct permutations $\pi, \sigma\in \Omega$, we let $hd(\pi, \sigma) = |\{ x\in \Omega: \pi(x) \ne \sigma(x) \}|$, called the Hamming…

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, the authors of a recent paper showed that the expected number of…

We consider partitions of a point set into two parts, and the lengths of the minimum spanning trees of the original set and of the two parts. If $w(P)$ denotes the length of a minimum spanning tree of $P$, we show that every set $P$ of $n…

We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\mathcal{P}(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive…

Normal approximations for descents and inversions of permutations of the set $\{1,2,...,n\}$ are well known. A number of sequences that occur in practice, such as the human genome and other genomes, contain many repeated elements. Motivated…

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…

The poset of permutations of [n] under Bruhat ordering is studied. We give nontrivial upper and lower bounds for the number of comparable pairs of permutations in both the weak and strong versions of this order. In light of numerical…

For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that…

Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, \ldots, s^k$ and $t^1, \ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci\'nski posed…

A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of…

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length n with k holes is a sequence of symbols $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the…

For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and…