如果一个排列可以被划分为 k 个单调子序列，则称其为 k-可覆盖的。Barber 猜想：对于任意给定的排列，如果每个长度为 \inom{k+2}{2} 的子序列都是 k-可覆盖的，那么该排列本身也是 k-可覆盖的。该猜想若成立，将是最优的。本文的目标是否定所有 k ≥ 3 情形下的这一猜想。事实上，我们证明了对任意 k，存在这样的排列：其每个长度至多为 (k/6)^{2.46} 的子序列都是 k-可覆盖的，但该排列本身却不是。
英文摘要：
A permutation is $k$-coverable if it can be partitioned into $k$ monotone subsequences. Barber conjectured that, for any given permutation, if every subsequence of length $k+2 \choose 2$ is $k$-coverable then the permutation itself is $k$-coverable. This conjecture, if true, would be best possible.
Our aim in this paper is to disprove this conjecture for all $k \ge 3$. In fact, we show that for any $k$ there are permutations such that every subsequence of length at most $(k/6)^{2.46}$ is $k$-coverable while the permutation itself is not.
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length $k-1$.
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $Ω_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random $k$-dimensional permutation of order $n$ is asymptotically almost surely $Θ_{k}\left(n^{\frac{k}{k+1}}\right)$.
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 $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all …
We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions.
We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permut…
One of the most classical results in Ramsey theory is the theorem of Erdős and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following natural question motivated by this result: Given integers $k$ and $n$ with $n \geq k^2+1$, how many monotone subsequences of length $k+1$ must every sequence of $n$ numbers contain? We answer this question precisely for all sufficiently large $k$ and $n \leq k^2 + c k^{3/2} / \log k$, where $c$ is some absolute positive constant.
Recently, Avgustinovich, Kitaev, and Taranenko defined five types of $(k, \ell)-$crucial permutations, which are maximal permutations that do not contain an increasing subsequence of length $k$ or a decreasing subsequence of length $\ell$. Further, Avgustinovich, Kitaev, and Taranenko began the enumeration of the $(k, \ell)-$crucial permutations of the minimal length and the next minimal length and the $(k, 3)-$crucial permutations of all lengths for each of the five types of $(k,\ell)-$crucial permutations. In this paper, we complete the enumeration that Avgustinovich, Kitaev, and Taranenko b…
We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $τ$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern $τ$ is monotone increasing or decreasing, or any pattern of length four.
A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $ρ$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any $r\ge0$, the largest union of $\lfloor r\sqrt{n}\rfloor$ de…
We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.
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 is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviati…
We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering an…
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 infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the correspond…
We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon [n] \to \mathbb{R}$, provided that $f$ is $\varepsilon$-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes $O(\log n)$ queries. This i…
We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
For a fixed irrational number $α$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = αi \mod 1$. Our primary result is that the boundary of the Schensted shape is approximated by a piecewise linear function with at most two slopes. This piecewise linear function is explicitly described in terms of the continued fraction expansion for $α$. Our results generalize those of Boyd and Steele, who studied longest monotone subsequences. Our proofs are based on a careful analysis of monotone sets in two-dimensional lattices.
A pattern $τ$ is a permutation, and an arithmetic occurrence of $τ$ in (another) permutation $π=π_1π_2...π_n$ is a subsequence $π_{i_1}π_{i_2}...π_{i_m}$ of $π$ that is order isomorphic to $τ$ where the numbers $i_1<i_2<...<i_m$ form an arithmetic progression. A permutation is $(k,\ell)$-crucial if it avoids arithmetically the patterns $12... k$ and $\ell(\ell-1)... 1$ but its extension to the right by any element does not avoid arithmetically these patterns. A $(k,\ell)$-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence of $12... k$ or $\ell(\el…
We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2…
Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of $Ω(\log N)$ points. Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k+1)-…
It is a classical fact that for any $\varepsilon > 0$, a random permutation of length $n = (1 + \varepsilon) k^2 / 4$ typically contains a monotone subsequence of length $k$. As a far-reaching generalization, Alon conjectured that a random permutation of this same length $n$ is typically $k$-universal, meaning that it simultaneously contains every pattern of length $k$. He also made the simple observation that for $n = O(k^2 \log k)$, a random length-$n$ permutation is typically $k$-universal. We make the first significant progress towards Alon's conjecture by showing that $n = 2000 k^2 \log \…
The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word $w$ fixes $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. In this paper, we ask for the existence of fixing words,…
Let $π$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $π$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $Ω(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showin…
We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations $τ$ (the `text') and $π$ (the `pattern'), and the goal is to decide whether $τ$ contains $π$ as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern $σ$; this restriction is known as Av($σ$)-PPM. It has been previously shown that Av($σ$)-PPM is polynomial for any $σ$ of size at most 3, …
