arXiv math/05025042005-02-23math.CO

On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia

M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott, M. Zabrocki

备注：: Submitted to Advances in Applied Mathematics
发表信息：: Advances in Applied Mathematics 36(2) Special Issue on Pattern Avoiding Permutations (2006) pages 96-105
DOI：: 10.1016/j.aam.2005.05.007

英文摘要：

We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Füredi-Hajnal and Stanley-Wilf conjectures in higher dimensions

In this paper we discuss analogs of Füredi-Hajnal and Stanley-Wilf conjectures for $t$-dimensional matrices with $t>2$.

2021-03-26math.CO

Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also …

2011-11-24math.CO

A short note on the Stanley-Wilf Conjecture for permutations on multisets

The concept of pattern avoidance respectively containment in permutations can be extended to permutations on multisets in a straightforward way. In this note we present a direct proof of the already known fact that the well-known Stanley-Wilf Conjecture, stating that the number of permutations avoiding a given pattern does not grow faster than exponentially, also holds for permutations on multisets.

2013-06-21math.CO

有根标号森林中模式的Stanley-Wilf极限

Stanley-Wilf Limits for Patterns in Rooted Labeled Forests

该研究基于Garg和Peng的工作，用解析方法证明了有根标号森林中规避单模式及部分模式集合时的Stanley-Wilf猜想类比结论，可计算对应极限的收敛下界，并提出了相关开放研究问题。

2023-10-04math.CO

The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence of patterns. Finally, we provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if $q$ is a layered pattern of length $k$, then $L(q)\geq (k-1)^2$ holds.

2004-05-13math.CO

Stanley-Wilf limits are typically exponential

For a permutation $π$, let $S_{n}(π)$ be the number of permutations on $n$ letters avoiding $π$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(π)= \lim_{n \to \infty} S_n(π)^{1/n}$ exists and is finite. Backed by numerical evidence, it has been conjectured by many researchers over the years that $L(π)=Θ(k^2)$ for every permutation $π$ on $k$ letters. We disprove this conjecture, showing that $L(π)=2^{k^{Θ(1)}}$ for almost all permutations $π$ on $k$ letters.

2013-10-31cs.DMmath.CO

更多相关论文收起更多论文

关于可块排 permutation 的 Stanley-Wilf 限制的增长率

On the growth rate of the Stanley-Wilf limit of blockable permutations

North Carolina School of Science and Mathematics（北卡罗来纳州科学数学学校）；Department of Mathematics, Duke University（杜克大学数学系）

本文研究了可块 permutation 的 Stanley-Wilf 限制的增长率，发现其在特定结构类中呈现多项式增长。

2025-12-27math.CO

关于Stanley-Wilf极限与Füredi-Hajnal极限之间的二次关系

On a Quadratic Relation Between Stanley-Wilf Limits and Füredi-Hajnal Limits

本文通过块收缩方法改进了Stanley-Wilf极限与Füredi-Hajnal极限之间的二次关系，得出了更精确的上界。

2026-07-14math.CO

Füredi-Hajnal 和 Stanley-Wilf 极限的典型增长

Typical growth of the Füredi-Hajnal and Stanley-Wilf limits

本文证明一致随机置换矩阵的 Füredi-Hajnal 和 Stanley-Wilf 极限以高概率至多为 exp(O(√k(log k)^{5/2}))，改进先前上界，并结合下界表明对数阶为 k^{1/2+o(1)}。

2026-10-05cs.DMmath.CO

Shape-Wilf-ordering of permutations of length 3

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, Bona has provided the only known up to now result of its type on ordering of permutations: |S_n(1342)|<|S_n(1234)|<|S_n(1324)| for n>6. We give a different proof of this result by ordering S_3 up to the stronger shape-Wilf-order: |S_Y(213)|<=|S_Y(123)|<=|S_Y(312)| for any Young diagram Y, derive as a consequence that |S_Y(k+2,k+1,k+3,tau)|<=|S_Y(k+1,k+2,k+3,tau…

2006-09-22math.CO

Subsequence containment by involutions

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in S_n which contain a given permutation τin S_k as a subsequence; this number depends on the patterns of the first j values of τfor 1<=j<=k. We then use this to define a partition of S_k, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation τ_1...τ_k is layered iff, for 1<=j<=k, the pattern of τ_1...τ_j is an involution. We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size $n$ w…

2004-09-02math.CO

避免模式的排列作为行走

Pattern Avoiding Permutations as Walks

University of Iceland（冰岛大学）

通过将排列视为有向图中的行走，研究1324模式的斯坦利-威尔极限，得出下界为10.418。

2025-12-22math.CO

On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation

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 that there is some constant c such that the mean value of this length is asymptotic to twice the square root of c times n and that the distribution of the length is tightly concentrated around its mean. We observe some apparent connections between c and the Stanley-Wilf limit of the class of permutations avoiding the given pattern.

2005-05-31math.CO

Asymptotics of Pattern Avoidance in the Permutation-Tuple and Klazar Set Partition Settings

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. One of the results of this paper extends work of Alweiss, and finds a classification for set partitions $π$ such that the number of set partitions of $[n]$ avoiding $π$ grows more slowly than $n^{cn}$ for all $c>0$. Several conjectures are proposed, and the related question of asymptotics of parallel ($k$-tuple) permutation pattern avoidance is considered and solved completely to within an exponential factor, generalizing Marcus and Tardos's 2004 proof of the Stanley-Wilf Conjecture.

2019-06-23math.CO

Pattern Avoidance in Ordered Set Partitions

In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n-1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type that exhib…

2013-03-24math.CO

Asymptotic enumeration of permutations avoiding generalized patterns

Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial resu…

2005-05-12math.CO

Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutations

The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given pattern $v$ exactly $r$ times. In addition, we study the asymptotic behavior of the random variable $X_n,$ the number of pattern occurrences in a random $n$-array word. The two topics are closely related through the identity $P(X_n=r) = $ $\frac{1}{k^n}f_r^v(k,n).$ In particular, we show that for any $r\geq 0,$ the Stanley-Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{…

2019-05-14math.COmath.PR

Better upper bounds on the Füredi-Hajnal limits of permutations

A binary matrix is a matrix with entries from the set $\{0,1\}$. We say that a binary matrix $A$ contains a binary matrix $S$ if $S$ can be obtained from $A$ by removal of some rows, some columns, and changing some $1$-entries to $0$-entries. If $A$ does not contain $S$, we say that $A$ avoids $S$. A $k$-permutation matrix $P$ is a binary $k \times k$ matrix with exactly one $1$-entry in every row and one $1$-entry in every column. The Füredi-Hajnal conjecture, proved by Marcus and Tardos, states that for every permutation matrix $P$, there is a constant $c_P$ such that for every $n \in \mathb…

2019-10-13cs.DMmath.CO

基于模式避免输入的优化

Optimization with pattern-avoiding input

本文研究了模式避免输入对优化问题复杂性的影响，证明了在动态最优猜想下的最优二叉搜索树的平均搜索成本为常数，并展示了该结论在其他算法问题中的广泛应用。

2025-11-24cs.DSmath.CO