arXiv 1007.46632010-07-27cs.DMmath.CO

The Multiple Permutation Problem and Some Conjectures

Zan Pan

备注：: LaTeX, 16 pages

英文摘要：

In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences' terms, our problem can be really difficult, which calls for novel approaches to work it out for any given pair $(m,n)$. Complete or partial solutions for $m=2, 3$ with smaller $n$'s are listed. Moreover, we have proved the necessary condition for $p(m,n) \neq 0$ and suggested an elegant asymptotic formula for $p(2,n)$. In addition, several challenging conjectures are provided, together with concise comments.

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

Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted per…

2022-09-25cs.DMmath.CO

A simplified disproof of Beck's three permutations conjecture and an application to root-mean-squared discrepancy

A $k$-permutation family on $n$ vertices is a set system consisting of the intervals of $k$ permutations of the integers $1$ through $n$. The discrepancy of a set system is the minimum over all red-blue vertex colorings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any $3$-permutation family is at most a constant independent of $n$. Here we give a simpler proof that Newman and Nikolov's sequence of $3$-permutation families has discrepancy $Ω(\log n)$. We also exh…

2018-11-27cs.DMmath.CO

A permutation code preserving a double Eulerian bistatistic

Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the Möbius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show t…

2016-06-25cs.DMmath.CO

A note on X-rays of permutations and a problem of Brualdi and Fritscher

The subject of this note is a challenging conjecture about X-rays of permutations which is a special case of a conjecture regarding Skolem sequences. In relation to this, Brualdi and Fritscher [Linear Algebra and its Applications, 2014] posed the following problem: Determine a bijection between extremal Skolem sets and binary Hankel X-rays of permutation matrices. We give such a bijection, along with some related observations.

2017-07-12cs.DMmath.CO

A counterexample to Beck's conjecture on the discrepancy of three permutations

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. Jozsef Beck conjectured (c. 1987) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3^k, we exhibit three permutations whose corresponding set system has discrepancy Omega(log(n)). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck's conjecture due to Spence…

2011-04-14cs.DM

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

当箭头模式与经典模式相遇时

When arrow patterns meet classical patterns

Chongqing University（重庆大学）

研究旨在弥合排列的循环表示法和单行表示法之间的结构鸿沟，通过引入箭头模式概念，解决了Archer和Laudone留下的三个关于箭头模式避免的猜想，包括枚举特定子类排列及给出不同证明。

2026-09-23cs.DMmath.CO

Algorithms for discovering and proving theorems about permutation patterns

We present an algorithm, called BiSC, that describes the patterns avoided by a given set of permutations. It automatically conjectures the statements of known theorems such as the descriptions of stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990), smooth (Lakshmibai and Sandhya 1990) and forest-like permutations (Bousquet-Melou and Butler 2007), and simsun permutations (Branden and Claesson 2011). The algorithm has also been used to discover new theorems and conjectures related to Young tableaux, Wilf-equivalences and sorting devices. We further give algorithms to p…

2012-11-29cs.DMcs.DScs.MS