Pattern Matching in the Cycle Structure of Permutations

The cycle enumerator of unimodal permutations

We give a generating function for the number of unimodal permutations with a given cycle structure.

Cycle up-down permutations

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the Euler numbers, and we study the distribution of some statistics on them, as well as on up-down permutations, on all permutations, and on a generalization of cycle-up-down permutations. The statistics include the number of cycles of even and odd length, the number of left-to-right minima, and the number of extreme elements.

On Connection between the Numbers of Permutations and Full Cycles with Some Restrictions on Positions and Up-Down Structure

We discuss both simple and more subtle connections between the numbers of permutations and full cycles with some restrictions,in particular, between the numbers of permutations and full cycles with prescribed up-down structure.

Blocks in cycles and k-commuting permutations

Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $β$ in terms of blocks in cycles in the decomposition of $β$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute …

Adjacent q-cycles in permutations

We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A generalization to more than one fixed length is also considered.

Recent results on permutations without short cycles

The density, denoted by $κ(n,r)$, of permutations having no cycles of length less than $r+1$ in a symmetric group $\mathrm{S}_n$ is explored. New asymptotic formulas for $κ(n,r)$ are obtained using the saddle-point method when $5\leq r< n$ and $n\to\infty$.

关于每个循环带权重$-1$的排列与D-排列的连分数的注记

A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle

该研究证明了每个循环带权重-1时，带多独立统计量的排列与D-排列的普通生成函数存在简洁连分数形式，并借助关联模2循环数与不动点、循环峰谷、交叉数的引理完成证明。

关于多维排列的一族通用循环

On a family of universal cycles for multi-dimensional permutations

本文提出一种基于贪心算法构造多维排列通用循环的方法，推广了 Gao 等人的排列 u-cycle 构造，并指出 d 维矩阵也存在 u-cycle。

Universal cycles for permutations

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.

A note on the total number of cycles of even and odd permutations

We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity: $$\sum_{i=1}^n c(n,i)\cdot i \cdot (-k)^{i-1} = (-1)^k k! (n-k-1)!,$$ where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.

On shortening u-cycles and u-words for permutations

This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for $n$-permutations exist of lengths $n!+(1-k)(n-1)$ for $k=0,1,\ldots,(n-2)!$.

具有模式受限循环形式的模式受限循环排列

Pattern-restricted cyclic permutations with a pattern-restricted cycle form

本文研究了避免单调递减排列的循环排列，在循环标准形式避免特定长度模式或所有循环旋转避免特定长度模式的情况下，分别通过连分数生成函数给出答案并枚举了相关情形。

按循环数枚举两个置换类

Enumerating two permutation classes by the number of cycles

本文按循环数对两个置换类 Av_n(312,4321) 和 Av_n(321,4123) 进行枚举，并针对若干统计量细化结果。

关于置换的对称张量幂的循环结构

On the cycle structure of the symmetric tensor power of permutations

本文研究置换σ的循环分解与其对称张量幂σ^{⊙k}的循环分解之间的关系，给出了计算不动点数量的公式，并在σ至多含有一个长度大于1的循环时给出了s-循环数量的公式。

关于多重集排列、停车函数及相关结构中的环

On Cycles in Multiset Permutations, Parking Functions, and Related Structures

本文研究多重集排列和停车函数中的环，通过代数和分析方法给出精确与渐近结果，并揭示终端闭包与循环点之间的意外对应关系。

Pattern-restricted permutations composed of 3-cycles

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n-1}$, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-known Motzkin numbers and Catalan numbers. The number of permutations composed of $n$ 3-cycles that avoid the pattern 321 …

Enumerating cycles in the graph of overlapping permutations

The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots a_{n+1}$ would connect the standardization of $a_1\cdots a_n$ to the standardization of $a_2\cdots a_{n+1}$. We examine properties of this graph to determine where directed cycles can exist, to count the number of directed $2$-cycles within the graph, and to enumerate the vertices that are contained within closed walks and directed cycles of more general lengt…

s-Overlap Cycles for Permutations

The goal of this paper is to solve Problem 481 from the list of research problems in the special issue of Discrete Mathematics dedicated to the Banff International Research Station workshop on "Generalizations of de Bruijn Cycles and Gray Codes" in 2004. Overlap cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally in 2010 by Godbole et al. In this paper we prove that s-overlap cycles for k-permutations of [n] exist for all k<n.

计数其乘积为具有受限循环长度的排列的循环对

Counting pairs of cycles whose product is a permutation with restricted cycle lengths

本文研究乘积循环长度受限时 $N$-循环对的计数，给出精确与渐近公式，并据此证明与恒等排列块转置距离至少为 $(n+1)/2$ 的排列数量具有较高下界。

仅含奇长度或仅含偶长度循环的排列的下降集分布

Descent set distribution for permutations with cycles of only odd or only even lengths

本文证明对称群中仅含奇长度循环与仅含偶长度循环的排列数等式在下降集层面的细化，并给出奇数阶情形的变体，方法基于特征标生成函数与高阶 Lie 特征标新恒等式。

Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics $(\sor, \cyc)$ and $(\inv, \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sor…

Eulerian polynomials, perfect matchings and Stirling permutations of the second kind

In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles.

Equidistributed statistics on matchings and permutations

We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization. As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our re…

在单行形式和循环形式中同时避免模式的循环排列

Cyclic permutations avoiding patterns in both one-line and cycle forms

本文研究长度为3的模式对(σ,τ)，枚举对称群中在单行记号避免σ且在循环记号避免τ的循环排列数量。

Two Enumerative Results on Cycles of Permutations

Answering a question of Bona, it is shown that for n>1 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,...,n} is 1/2 if n is odd and 1/2 - 2/(n-1){n+2) if n is even. Another result concerns the generating function P_h(q) for the number of cycles of the product (1,2,...,n)w, where w ranges over all permutations of 1,2,...,n of cycle type h. A formula is obtained for P_h(q) from which it is proved that the zeros of P_h(q) have real part 0.

Bijective enumeration of some colored permutations given by the product of two long cycles

Let $γ_n$ be the permutation on $n$ symbols defined by $γ_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $β$ of $n$ in which the numbers from 1 to $n$ are colored with $p$ colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that $γ_n β^{-1}$ is a long cycle is given by the very simple ratio $\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportio…

Separation probabilities for products of permutations

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles.

循环交替排列的连分数

Continued fractions for cycle-alternating permutations

该研究针对循环交替排列的计数多元多项式得到Stieltjes型连分数，引入其推广结构交替Laguerre有向图并给出对应计数多项式的指数生成函数，还从图的角度解释了相关连分数对应的两类Rogers矩阵。

排列的缩短通用循环

Shortened universal cycles for permutations

本文通过构造性证明验证了 Kitaev 等人关于利用不可比较元素将排列通用循环缩短至 $n!-i(n-1)$ 的猜想，并提出一种新的排列通用循环构造方法。

Number of cycles in the graph of 312-avoiding permutations

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation $π= π_{1} π_{2} ... π_{n+1}$ there is a directed edge from the standardization of $π_{1} π_{2} ... π_{n}$ to the standardization of $π_{2} π_{3} ... π_{n+1}$. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.

Random and exhaustive generation of permutations and cycles

In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well known. We consider both methods in a unified way, and discuss their relation with exhaustive generation methods. We analyse several random variables associated with the algorithms and find their grand probability generating functions, which gives easy access to moments and limit laws.

On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs

We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input. We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.

Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This …

富集的循环结构与置换的根

Enriched Cycle Structures and Roots of Permutations

本文通过定义富集置换，揭示了 $r$-正则置换与富集 $r$-循环置换之间的双射关系，从而证明了 $r$-th 根概率的单调性质，并回答了 Bóna-McLennan-White 的问题。

Nestings of Matchings and Permutations and North Steps in PDSAWs

We present a simple bijective proof of the fact that matchings of [2n] with N nestings are equinumerous to partially directed self avoiding walks confined to the symmetric wedge defined by y=+-x, with n east steps and N north steps. A very similar construction connects permutations with N nestings and PDSAWs remaining below the x-axis, again with N north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.

Enumeration of snakes and cycle-alternating permutations

Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cycle-alternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain the generating functions,…

Cycles and patterns in permutations

We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating functions for inversions, ascents, descents, 123s, valleys, 1'-2-1s; closed forms forms for avoidance of peaks, 2-3-1s, 1-2-3s, 2'-1-2s and 1'-2-1s; bijective proofs of Wilf-equivalences. We also derive some results about standard pattern occurrence, such as continued fractions for the generating functions for occurrences of valleys and the pattern 123. The…

2-相交排列族和完美匹配族的唯一性

Uniqueness for 2-Intersecting Families of Permutations and Perfect Matchings

本文刻画了所有n≥2时{1,2,…,n}的最大2-相交排列族和完全图K_{2n}的最大2-相交完美匹配族。

平衡加权与置换匹配在保序映射幺半群中的应用

Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings

本文给出有限半群组合D类存在置换匹配的充要条件，即结构矩阵存在平衡加权，并应用此条件证明n元链上保序自映射幺半群具有置换匹配。

Combinatorially refine a Zagier-Stanley result on products of permutations

In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a formula of the number of pairs of long cycles whose product has $k$ cycles independently obtained by Zagier and Stanley relying on group characters, and was previously obtained by Féray and Vassilieva by counting some colored permutations first and then relying on some algebraic computations in the ring of symmetric functions. Our approach here is simpler and…

Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.

线性化置换多项式的循环结构与奇偶性

Cycle structures and parities of linearized permutation polynomials

本文刻画线性化置换多项式的循环结构与奇偶性，为Mullen和Vaughan1988年的问题提供迪克森矩阵形式表述，推导奇偶性判据并给出判定算法，应用于编码、密码等领域。

欧拉型多项式与匹配及匹配排列

Eulerian-type polynomials over matchings and matching permutations

本文研究了匹配与排列之间的联系，揭示了匹配中四元统计量与排列中相应统计量的对应关系，并通过引入匹配排列，扩展了五变量邻居多项式与二阶欧拉多项式的关系。

Minimal transitive factorizations of permutations into cycles

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc. with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain "descent structure" and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to count factorizations of permutations with one or two parts, extending earlier w…

Independent set and matching permutations

Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $π$ of $\{1, \ldots, m\}$ is an {\em independent set permutation} of $G$ if $$ a_{π(1)}(G) \leq a_{π(2)}(G) \leq \cdots \leq a_{π(m)}(G) $$ where $a_k(G)$ is the number of independent sets of size $k$ in $G$. In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $α(G)=m$, i.e. with largest independent set having size $m$. They raised the question of determining, for each $m$, the smallest number $f(m)$ such that every permut…

RSK-Complete Cycle Decompositions

We characterize the class of cycle decompositions that can achieve all Young tableau shapes (except the trivial ones with a single row or a single column) under the Robinson--Schensted--Knuth (RSK) correspondence, a property that we call RSK-completeness. We prove that for even $n$, cyclic permutations comprise the only fixed cycle decomposition that is RSK-complete. For odd $n$, cyclic permutations and almost cyclic permutations which have a cycle of length $n-1$ are the only RSK-complete cycle decompositions.

Cycle structure of random permutations with cycle weights

We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.

The exact bound for the Erdős-Ko-Rado theorem for $t$-cycle-intersecting permutations

In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erdős-Ko-Rado Theorem to show that if $n \geq 2t+1$, then any pairwise $t$-cycle-intersecting family of permutations has cardinality less than or equal to $(n-t)!$. Furthermore, the only families attaining this size are the stabilizers of $t$ points, that is, families consisting of all permutations having $t$ 1-cycles in common. This is a strengthening of a previous result of Ku and Renshaw and supports a recent conjecture by Ellis, Friedgut and Pilpel concerning the corresponding bound for $t$-in…

循环置换图与置换snarks的生成

Generation of Cycle Permutation Graphs and Permutation Snarks

本文提出算法生成所有非同构的循环置换图，包括非哈密顿的循环置换图和置换snarks，提高了之前结果的上限，并提供了关于置换snarks的下界和猜想的证据。

The cycle polynomial of a permutation group

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $Ω$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbit…

平衡格雷码用于排列与结合多面体的彩虹环

Balanced Gray Codes for Permutations and Rainbow Cycles for Associahedra

本研究解决了排列的平衡转置格雷码构造问题，并推广了彩虹环概念，为结合多面体和排列多面体提供了新的彩虹环构造，同时补充了偶数情况下的平衡格雷码。

On the constructions of $n$-cycle permutations

Any permutation polynomial is an $ n $-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for $ n $-cycle permutations, which mainly are of the form $ x^rh(x^s) $. We then propose unified constructing methods including recursive ways and a cyclotomic way for $ n $-cycle permutations of such form. We demonstrate our approaches by c…

Integer sequences and k-commuting permutations

Let $β$ be any permutation on $n$ symbols and let $c(k, β)$ be the number of permutations that $k$-commute with $β$. The cycle type of a permutation $β$ is a vector $(c_1, \dots, c_n)$ such that $β$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, β)$, for some cycle types. We also express these formulas in terms of integer sequences as given in "The On-line Encyclopedia of Integer Sequences" (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.

关于轮换排列模式避免的一个猜想

On a conjecture about pattern avoidance of cycle permutations

本文研究轮换排列的模式避免问题，证明了对称群中单行形式避免2431且所有轮换形式避免1324的轮换排列数量为第(n-1)个Pell数，肯定回答了Archer等人的猜想。