Related papers: Longest monotone subsequences and rare regions of …

For a fixed irrational number $\alpha$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = \alpha i \mod 1$. Our primary result is that the boundary of the Schensted…

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random…

Let T_k^m={\sigma \in S_k | \sigma_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any \tau in T_k^1 (or any \tau in T_k^k), the number of…

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$…

We study a class of random permutons which can be constructed from a pair of space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by…

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)…

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…

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets…

The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits sqrt[n](|S_n(tau)|) and further works suggest asymptotic ordering of layered versus monotone patterns. Yet,…

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…

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal…

The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids…

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set…

Multidimensional permutations, or $d$-permutations, are represented by their diagrams on $[n]^d$ such that there exists exactly one point per hyperplane $x_i$ that satisfies $x_i= j$ for $i \in [d]$ and $j \in [n]$. Bonichon and Morel…

Sequence pattern avoidance is a central topic in combinatorics. A sequence $s$ contains a sequence $u$ if some subsequence of $s$ can be changed into $u$ by a one-to-one renaming of its letters. If $s$ does not contain $u$, then $s$ avoids…

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…

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…

We consider pattern containment and avoidance with a very tight definition that was used first by Riordan more than 60 years ago. Using this definition, we prove the monotone pattern is easier to avoid than almost any other pattern of the…

The study of pattern avoidance in linear permutations has been an active area of research for almost half a century now, starting with the work of Knuth in 1973. More recently, the question of pattern avoidance in circular permutations has…

We introduce a permutation analogue of the celebrated Szemeredi Regularity Lemma, and derive a number of consequences. This tool allows us to provide a structural description of permutations which avoid a specified pattern, a result that…