How often is a permutation an n'th power?

排列幂中的下降

Descents in powers of permutations

本文研究排列幂的下降数分布，枚举了平方按下降数分类的 Grassmannian 排列、平方恰有一个下降的情形及平方或立方达最大下降数的排列。

Stack-Sorting Preimages of Permutation Classes

We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$. First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(π)|$ for any permutation $π$, answering a question of Bousquet-Mélou. We then enumerate permutations in various sets of the form $s^{-1}(\text{Av}(τ^{(1)},\ldots,τ^{(r)}))$, where $\text{Av}(τ^{(1)},\ldots,τ^{(r)})$ is the set of permutations avoiding the patterns $τ^{(1)},\ldots,τ^{(r)}$. These preimage sets often turn out to be permutation classes themselves, so the current paper represents a new approach, based on …

排列的幂与合成中的模式避免

Pattern avoidance in compositions and powers of permutations

本文定义正整数合成的模式避免概念，并利用它枚举避免链 (312,321:σ) 的长度为 n 的排列，其中 σ 为任意模式，同时枚举避免链 (312,4321:σ) 的排列，其中 σ 为 S_3 中的模式。

The absence of a pattern and the number of occurrences of another

Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern $q$ in permutations that avoid another given pattern $r$. In some cases, we find the pattern that occurs least often, (resp. most often) in all $r$-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.

Intersecting families of sets and permutations: a survey

A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest $t$-intersecting sub-family of a given family $\mathcal{F}$. We give a survey of known results, conjectures and open problems for various important families $\mathcal{F}$, namely, power sets, levels of power sets, hereditary families, families of signed sets, families of labeled sets, and families of permutations. We also provide some extensions and consequences of kno…

Plethystic exponential calculus and characteristic polynomials of permutations

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of symmetric products of real/algebraic tori. The proof uses formal power series expansions of plethystic exponentials, and has been motivated by some recent applications of these combinatorial tools in supersymmetric gauge and string theories. Since the methods are elementary, we tried to be self-contained, and relate to other topics such as the q-binoomial theorem,…

The $\hat W$-orbit of $ρ$, Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z

Let an affine Weyl group $\hat W$ act as a group of affine transformations on a real vector space V. We analyze the $\hat W$-orbit of a regular element in V and deduce applications to Kostant's formula for powers of the Euler product and to the representations of $\hat W$ as permutations of the integers.

Symmetric powers of permutation representations of finite groups and primitive colorings on polyhedrons

In this paper, we define a set which has a finite group action and is generated by a finite color set, a set which has a finite group action, and a subset of the set of non negative integers. we state its properties to apply one of solution of the following two problems, respectively. First, we calculate the generating function of the character of symmetric powers of permutation representation associated with a set which has a finite group action. Second, we calculate the number of primitive colorings on some objects of polyhedrons. It is a generalization of the calculation of the number of pr…

排列能有多平衡？

How Balanced Can Permutations Be?

本文研究排列的 $k$-balanced 性质，显式构造了 $k\le3$ 的平衡排列，并证明 $k\ge4$ 时不存在此类排列且给出了距离下界，在 $k=4$ 时构造匹配下界。

Operators of equivalent sorting power and related Wilf-equivalences

We study sorting operators $\mathbf{A}$ on permutations that are obtained composing Knuth's stack sorting operator $\mathbf{S}$ and the reversal operator $\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we provide a size-preserving bijection between the set of permutations sorted by $\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ \mathbf{R} \circ \mathbf{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this …

On super-strong Wilf equivalence classes of permutations

Super-strong (elsewhere referred to as strong) Wilf equivalence is a type of Wilf equivalence on words that was introduced by Kitaev et al. in 2009. We provide a necessary and sufficient condition for two permutations in $n$ letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in (Kitaev et al., 2009) that the cardinality of each Wilf equivalence class is a po…

算术函数与置换幂的不动点

Arithmetic functions and fixed points of powers of permutations

本文研究置换幂的不动点计数函数如何确定置换的共轭类，构造了计算共轭类的算法，并刻画了可作为不动点计数函数的算术函数。

A formal identity involving commuting triples of permutations

We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.

On the number of mth roots of permutations

Let m be a fixed positive integer. It is well-known that a permutation $σ$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of σ. Let p_m(n) be the probability that a random n-permutation has an mth root. We also include a proof that p_m(jq)=p_m(jq+1)=... =p_m(jq+(q-1)) where j=0,1,... and m is a power of prime q.

A recursive bijective approach to counting permutations containing 3-letter patterns

We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller multiset, and involves a consideration of separate cases according to how the patterns overlap. Specifically, we use the method (i) to provide combinatorial proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123 pattern, (ii) to express the number of n-permutations c…

Determination of a Type of Permutation Trinomials over Finite Fields, II

Let $q$ be a prime power. We determine all permutation trinomials of $\Bbb F_{q^2}$ of the form $ax+bx^q+x^{2q-1}\in\Bbb F_{q^2}[x]$. The subclass of such permutation trinomials of $\Bbb F_{q^2}$ with $a,b\in\Bbb F_q$ was determined in a recent paper by the author.

Compositions of Functions and Permutations Specified by Minimal Reaction Systems

This paper studies mathematical properties of reaction systems that was introduced by Enrenfeucht and Rozenberg as computational models inspired by biochemical reaction in the living cells. In particular, we continue the study on the generative power of functions specified by minimal reaction systems under composition initiated by Salomaa. Allowing degenerate reaction systems, functions specified by minimal reaction systems over a quarternary alphabet that are permutations generate the alternating group on the power set of the background set.

竞争增长过程中的排列与球入箱模型

Permutations in competing growth processes and balls-in-bins

研究N个独立递增过程在随机等待时间下的竞争增长，证明在无领导者时所有排名排列几乎必然无限频繁出现，并验证了Spencer关于带反馈球入箱模型的猜想。

A family of permutation trinomials in $\mathbb{F}_{q^2}$

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary.

有限域上提供对合置换的多项式集合

Collection of Polynomials over Finite Fields Providing Involutary Permutations

本文针对满足特定同余条件的奇素数幂，构造了一类有限域上的置换多项式集合，证明它们均能生成具有特定不动点数量的对合置换，并明确给出了三项式与六项式的数量分布。

A Class of Permutation Binomials over Finite Fields

Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv 1\pmod 4$; (ii) $t=-3$, $q\equiv \pm1\pmod{12}$; (iii) $t=3$, $q\equiv -1\pmod 6$. We confirm this conjecture in the present paper.

素数幂阶拉丁方的代数方法：基于局部置换多项式

An algebraic approach to Latin squares of prime power order by local permutation polynomials

本文通过局部置换多项式将素数幂阶拉丁方系数与代数集零点等同，提出统一代数描述方法，并处理约化与同位拉丁方，在q=4,5上构造验证，最后联系完全映射与横截。

New Lower Bounds for Permutation Arrays Using Contraction

A permutation array $A$ is a set of permutations on a finite set $Ω$, say of size $n$. Given distinct permutations $π, σ\in Ω$, we let $hd(π, σ) = |\{ x\in Ω: π(x) \ne σ(x) \}|$, called the Hamming distance between $π$ and $σ$. Now let $hd(A) =$ min$\{ hd(π, σ): π, σ\in A \}$. For positive integers $n$ and $d$ with $d\le n$, we let $M(n,d)$ be the maximum number of permutations in any array $A$ satisfying $hd(A) \geq d$. There is an extensive literature on the function $M(n,d)$, motivated in part by suggested applications to error correcting codes for message transmission over power lines. A b…

有限特征下一类置换有理函数的显式确定

Explicit determination of a class of permutation rational functions in any characteristic

本文利用低次置换有理函数的分类结果及其几何性质，研究诱导乘法子群mu_q+1置换的有理函数，明确描述了小次置换有理函数的一类广泛类别的显式结构，并应用这些发现确定了由度3有理函数诱导的许多F_q^2上的置换四次多项式。