On a cyclic inequality with exponents and permutations, and its Shapiro-type analogue

On cyclic Schur-positive sets of permutation

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. T…

Permutations over cyclic groups

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements $a_1,...,a_m$ of the cyclic group of order $m$, there is a permutation $π$ such that $1a_{π(1)}+...+ma_{π(m)}=0$.

Cyclic sieving phenomenon on annular noncrossing permutations

We show an instance of the cyclic sieving phenomenon on annular noncrossing permutations with given cycle types. We define annular $q$-Kreweras numbers, annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are polynomials in $q$. We then show that these polynomials exhibit the cyclic sieving phenomenon on annular noncrossing permutations. We also show that a sum of annular $q$-Kreweras numbers becomes an annular $q$-Narayana number and a sum of $q$-Narayana numbers becomes an annular $q$-Catalan number.

Descent sets of cyclic permutations

We present a bijection between cyclic permutations of {1,2,...,n+1} and permutations of {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. This non-trivial bijection involves a Foata-like transformation on the cyclic notation of the permutation, followed by certain conjugations. We also give an alternate derivation of the consequent result about the equidistribution of descent sets using work of Gessel and Reutenauer. Finally, we prove a conjecture of the author in [SIAM J. Discrete Math. 23 (2009), 765-786] and a conjecture of Eriksen, Freij and …

Vincular Pattern Avoidance on Cyclic Permutations

Pattern avoidance for permutations has been extensively studied, and has been generalized to vincular patterns, where certain elements can be required to be adjacent. In addition, cyclic permutations, i.e., permutations written in a circle rather than a line, have been frequently studied, including in the context of pattern avoidance. We investigate vincular pattern avoidance on cyclic permutations. In particular, we enumerate many avoidance classes of sets of vincular patterns of length 3, including a complete enumeration for all single patterns of length 3. Further, several of the avoidance …

Cyclic Pattern Containment and Avoidance

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdos-Szekeres Theorem that any linear permutation of length mn+1 must contain either the increasing pattern of length m+1 or the decreasing pattern of length n+1. We …

关于避免(132,213)的循环排列增长率的一个下界

A lower bound on the growth rate of $(132,213)$-avoiding cyclic permutations

研究给出避免(132,213)的循环排列增长率的首个非平凡下界。构建新约简过程，迭代可判断是否为循环排列，反向能构建此类排列。还得出奇偶大小类双射及受限层数排列精确计数等结果。

Asymptotic analysis of the cyclic structure of permutations

This is my dissertation. Its research object is a symmetric group of permutations acting on a finite set. The density of permutations with a given cycle structure pattern is explored when the group order tends to infinity. New and sharper asymptotic formulas are obtained. The latter are applied in approximations of the cycle vector distribution of a random permutation. The saddle-point method, Lagrange-Bürmann inversion formula, Laplace transformations, and other techniques of complex analysis are applied.

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。

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。

避免长度为三的模式对的循环排列

Cyclic permutations avoiding pairs of patterns of length three

本文完成了避免两个长度为三的模式的循环排列的枚举，给出了除一对外的所有显式公式，并证明了避免某无限族中单个模式的循环排列数量的增长率下界。

The CLLC conjecture holds for cyclic outer permutations

Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function counting cycles of some permutation compositions. In this paper, we confirm the CLLC conjecture for cyclic permutations, with the aid of Hultman numbers and by applying the Hermite--Biehler theorem on the generating function of Stirling numbers of the first kind. This leads to a further conjecture that every local genus polynomial is real-rooted.

Exact and asymptotic enumeration of cyclic permutations according to descent set

Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given descent set). We then use this formula to show that, for almost all sets $I \subseteq [n-1]$, the fraction of size-$n$ permutations with descent set $I$ which are $n$-cycles is asymptotically $1/n$. As a special case, we recover a result of Stanley for alternating cycles. We also use our formula to count the cycles that do not have two consecutive descents.

B型与D型循环置换的下降集

Descent sets of cyclic permutations in types B and D

本研究针对B型带号对称群构造了对应Elizalde双射的函数Φ，证明其在D型循环带号置换及补集上的限制为双射，且可退化为原双射，还得到了循环带号置换的下降与标志主指标统计量的渐近正态性。

Han's Bijection via Permutation Codes

We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $π$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $σ$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Björner and Wachs.

关于循环排列模式回避问题的研究

On a question about pattern avoidance of cyclic permutations

本文通过分析循环形式结构和应用Dilworth定理，解决了循环排列中避免特定模式τ=1432的问题。

Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n,k). This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine the eigenvalues of A(n,k) for small values of k. Finally, we show that any eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n…

不可约循环码的排列自同构群

The permutation automorphism groups of irreducible cyclic codes

本文通过有限简单群分类，对不可约循环码的排列自同构群进行了完整分类，证明除少数例外外，非退化不可约循环码均为标准，且为Schmidt-White猜想和Berger-Charpin猜想提供了证据。

关于循环排列模式回避的两个猜想

On two conjectures about pattern avoidance of cyclic permutations

本文研究循环排列在单行形式与标准循环形式中同时回避模式的问题，通过分析对称群中满足双重回避条件的循环排列集合，解决了 Archer 等人提出的两个猜想。

超八面体群的有符号图与有符号循环

Signed Graphs and Signed Cycles of Hyperoctahedral Groups

本文将Dénes关于图边序与全循环置换的经典结果推广到有符号图和超八面体群，给出了有符号图存在偶或奇全循环边序的充要条件，并统计了奇全循环置换的最少对换表示数目。

循环码的置换自同构群 I：分圆结合方案

Permutation automorphism groups of cyclic codes I: cyclotomic association schemes

本文研究循环码的置换自同构群，建立循环码与分圆结合方案的联系，证明 q-仿射群对某些码长不适用，并用算术条件刻画这些码长，其密度为零。

Permutation Polytopes of Cyclic Groups

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.

Cyclic Shuffle Compatibility

Consider a permutation p to be any finite list of distinct positive integers. A statistic is a function St whose domain is all permutations. Let S(p,q) be the set of shuffles of two disjoint permutations p and q. We say that St is shuffle compatible if the distribution of St over S(p,q) depends only on St(p), St(q), and the lengths of p and q. This notion is implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. One of the places where shuffles are useful is in describing the product in the algebra of quasisymmetric functions. Recently Adin, Gessel, R…

解决Schwartz二次曲折数猜想

Asymptotics of the Meandering Number of a Cyclic Permutation

通过定义循环排列的曲折数，证明其最大值在n的二次函数范围内，解决了Schwartz关于拓扑推销员问题的猜想，并构造了从线性到二次增长率的连续族。

渗流集合与排列的operad

Percolating sets and the operad of permutations

本文利用排列的operad重新解释Shapiro-Stephens关于渗流排列矩阵的刻画，并研究其理想、子operad与2-邻域渗流性质的关系，同时给出排列operad的一种表示。

On a conjecture concerning the shuffle-compatible permutation statistics

The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of shuffle-compatibility property…

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.

置换群中相对闭合的子群及其 cyclic 正则正规子群

Relatively closed subgroups of permutation groups with a cyclic regular normal subgroup

本文研究了具有 cyclic 正则正规子群的置换群中闭合子群的结构，通过分类最小非平凡一维仿射关联方案，扩展了最近的Muzychuk对一维仿射秩3图的分类结果。

On cyclic descents for tableaux

The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of {\em cyclic descent set} for permutations, and Rhoades introduced such a notion for SYT --- but only for rectangular shapes. In this work we define {\em cyclic extensions} of descent sets in a general context, and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials, providing a new interpretation of certain Gromov-Witten invariants.

通过循环 shuffle 代数研究循环 shuffle 相容性

Cyclic shuffle-compatibility via cyclic shuffle algebras

本文引入循环 shuffle 代数，并以循环拟对称函数替代拟对称函数，建立循环 shuffle 相容性的代数框架，进而刻画多种循环排列统计量并给出代数证明。

基于连分数组合学的循环筛分现象

Cyclic sieving phenomena via combinatorics of continued fractions

本文利用连分数恒等式研究排列、集合划分、完美匹配和D-排列上的循环筛分现象，重新证明并推广Corteel对合相关结果，证明两个猜想，并构造新的Genocchi-Corteel对合。

Descents of $λ$-unimodal cycles in a character formula

We prove an identity conjectured by Adin and Roichman involving the descent set of $λ$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas have previously been proven algebraically. In this paper, we present a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.

Intersection density of transitive groups with cyclic point stabilizers

For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $\mathcal{F}$ runs over all intersecting sets in $G$. If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the Erdős-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive…

Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Taking Shapiro's cyclic sums $\sum_{i=1}^n x_i/(x_{i+1}+x_{i+2})$ (assuming index addition mod $n$) as a starting point, we introduce a broader class of cyclic sums, called generalized Shapiro-Diananda sums, where the denominators are $p$-th order power means of the sets $\{x_{i+j_1},\dots,x_{i+j_k}\}$ with fixed distinct integers $j_1,\dots,j_k$ and $1\leq i\leq n$. Generalizing further, we replace the set of arguments of the power mean in the $i$-th denominator by an arbitrary nonempty subset of $\{1,\dots,n\}$ interpreted as the set of out-neighbors of the node number $i$ in a directed grap…

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation πof an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (π(a),π(b),π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mo…

Enumerations of Universal Cycles for $k$-Permutations

Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.

图的上可嵌入性与边关联的对换乘积

Upper Embeddability of Graphs and Products of Transpositions Associated with Edges

本文研究图的边序如何决定由边关联对换乘积得到的顶点置换，利用图嵌入刻画了存在边序使该置换为完全循环置换的图，并给出 Eden 关于恒等置换问题的一个反例。

Hopping from Chebyshev polynomials to permutation statistics

We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the $(-1)$-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic …

On rotated Schur-positive sets

The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of th…

Cyclic permutations realized by signed shifts

The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain cyclic permutation associated to the pattern. Signed shifts are an important family of one-dimensional dynamical systems that includes shift maps and the tent map as particular cases. Defined as a function on the set of infinite words on a finite alphabet, a signed shift deletes the first letter and, depending on …

排列上的循环筛选——FindStat数据库中映射与统计量的分析

Cyclic sieving on permutations -- an analysis of maps and statistics in the FindStat database

本研究借助SageMath检索FindStat数据库，系统分析排列统计量与双射映射的循环筛选现象，新证明34个CSP实例、提出3个猜想，还发现同态性作用与CSP候选并无必然对应关系。

On Volumes of Permutation Polytopes

This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.

项链、子集和与循环排列

Necklaces, subset sums, and cyclic permutations

本文探讨了子集和与项链之间的关系，引入参数r并研究其周期性条件，同时考虑子集大小k，揭示了项链中1的数量与子集和的关系。

Schurity of S-rings over a cyclic group and generalized wreath product of permutation groups

The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group $G$ and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that $G$ is a Schur group whenever the total number $Ω(n)$ of prime factors of the integer $n=|G|$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $Ω(n)=4$. The latter result implies in particular that if $n=p^3q$ …

Higher Lie characters and cyclic descent extension on conjugacy classes

A now-classical cyclic extension of the descent set of a permutation has been introduced by Klyachko and Cellini. Following a recent axiomatic approach to this notion, it is natural to ask which sets of permutations admit such an extension. The main result of this paper is a complete answer in the case of conjucay classes of permutations. It is shown that the conjugacy class of cycle type $λ$ has such an extension if and only if $λ$ is not of the form $(r^s)$ for some square-free $r$. The proof involves a detailed study of hook constituents in higher Lie characters.

On abelian-by-cyclic Moufang loops

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if …

通过相邻交换 conjugate 全周期：直径和排序时间

Conjugating full cycles by adjacent transpositions: diameter and sorting time

研究通过相邻交换 conjugate 循环排列的步骤数上限和 Schreier 图直径，探讨排序时间与直径的关系。

A Cyclic Analogue of Stanley's Shuffle Theorem

We introduce the cyclic major index of a cycle permutation and give a bivariate analogue of enumerative formula for the cyclic shuffles with a given cyclic descent numbers due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's Shuffle Theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.

有限域乘法子群中带多项式值的大子集的循环排列

Cyclic permutations of large subsets with polynomial values in multiplicative subgroups of finite fields

该研究针对有限域上的多项式与乘法子群，建立了大子集存在满足相邻元素多项式值属于k次幂集合的循环排列的阈值，并给出该阈值的上下界。

循环置换：度与组合类型

Cyclic Permutations: Degrees and Combinatorial Types

本文枚举对称群中n-循环按度的计数，通过动力系统周期轨道研究共轭类计数，并证明随机n-循环的度分布渐近正态。

Eulerian quasisymmetric functions and cyclic sieving

It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed cycle type and fixed number of excedances provides an instance of the cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.

Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons

We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.

乘法和指数正交映射的循环群变体

Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups

研究了循环群中乘法和指数正交映射的存在性与数量，证明乘法正交映射在n>2时不存在，而指数正交映射在n为两倍素数且p-1平方自由时存在并估计其数量。

On products of permutations with the most uncontaminated cycles by designated labels

There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of Bóna. This paper is concerned with a problem in that direction. Let $D$ be a permutation on the set $[n]=\{1,2,\ldots, n\}$ and $E\subset [n]$. Suppose the maximum possible number of cycles uncontaminated by the $E$-labels in the product of $D$ and a cyclic permutation on $[n]$ is $θ$ (depending on $D$ and $E$). We prove that for arbitrary $D$ and $E$ with few exceptions, the number of cyclic permutations $γ$ such that $D\circ γ$ has exactl…