The variance of the Stirling cycle numbers

Colorful Proofs of the Generating Formulas for Signed and Unsigned Stirling Numbers of the First Kind

We describe proofs of the standard generating formulas for unsigned and signed Stirling numbers of the first kind that follow from a natural combinatorial interpretation based on cycle-colored permutations.

关于特定图家族的图的r-斯特林数第一类

On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families

本文研究特定图家族的r-斯特林数第一类，推导其闭式表达式和递推关系，并分析循环分布的统计特性。

Stirling numbers of forests and cycles

For a graph $G$ and a positive integer $k$, the {\em graphical Stirling number} $S(G,k)$ is the number of partitions of the vertex set of $G$ into $k$ non-empty independent sets. Equivalently it is the number of proper colorings of $G$ that use exactly $k$ colors, with two colorings identified if they differ only on the names of the colors. If $G$ is the empty graph on $n$ vertices then $S(G,k)$ reduces to $S(n,k)$, the familiar Stirling number of the second kind. In this note we first consider Stirling numbers of forests. We show that if $(F^{c(n)}_n)_{n\geq 0}$ is any sequence of forests wit…

Mixed coloured permutations

In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, mixed Stirling numbers of the first kind. In this comprehensive study we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.

Some identities on degenerate r-stirling numbers via boson operators

Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal orde…

Stirling排列上的对称分解与Euler-Stirling统计量

Symmetric decompositions and Euler-Stirling statistics on Stirling permutations

本文研究Stirling排列上的Euler-Stirling统计量，通过文法变换与参数化建立$(p,q)$-Eulerian多项式和$(α,β)$-Eulerian多项式的联系，并给出多种对称分解及双$γ$-正性结果。

Linked Partitions and Linked Cycles

The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the n-th large Schroder number $r_n$, which counts the number of Schroder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including incr…

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…

Restricted Stirling and Lah number matrices and their inverses

Given $R \subseteq \mathbb{N}$ let ${n \brace k}_R$, ${n \brack k}_R$, and $L(n,k)_R$ be the number of ways of partitioning the set $[n]$ into $k$ non-empty subsets, cycles and lists, respectively, with each block having cardinality in $R$. We refer to these as the $R$-restricted Stirling numbers of the second and first kind and the $R$-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are ${n \brace k} = {n \brace k}_{\mathbb{N}}$, ${n \brack k} = {n \brack k}_{\mathbb{N}} $ and $L(n,k) = L(n,k)_{\mathbb{N}}$, res…

Extrema Property of the $k$-Ranking of Directed Paths and Cycles

A $k$-ranking of a directed graph $G$ is a labeling of the vertex set of $G$ with $k$ positive integers such that every directed path connecting two vertices with the same label includes a vertex with a larger label in between. The rank number of $G$ is defined to be the smallest $k$ such that $G$ has a $k$-ranking. We find the largest possible directed graph that can be obtained from a directed path or a directed cycle by attaching new edges to the vertices such that the new graphs have the same rank number as the original graphs. The adjacency matrix of the resulting graph is embedded in the…

Combinatorial and Arithmetical Properties of the Restricted and Associated Bell and Factorial Numbers

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus of the present article is the analysis of combinatorial and arithmetical properties of them. The results include several combinatorial identities and recurrences as well as some properties of their $p$-adic valuations.

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…

An Erd{\H o}s-Ko-Rado theorem for permutations with fixed number of cycles

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e., \[ S_{n,k} = \{π\in S_{n}: π= c_{1}c_{2} \cdots c_{k}\},\] where $c_1,c_2,\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is given by $\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\mathcal{A} \subseteq S_{n,k}$ is said to be $t$-{\em intersecting} if any two elements of $\mathcal{A}$ have at least $t$ com…

An Aearated Triangular Array of Integers

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when $n-k$ is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this extension produces a {\it second kind} of such integers which may be considered as a new generalization of the Genocchi nu…

Smirnov词与完全多部图中哈密顿圈的双射

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

建立平衡字母重数的Smirnov词与完全m部图K_{n,n,...,n}中哈密顿路径的双射，导出哈密顿圈数的闭式容斥公式，并推广到非均匀情形K_{n1,n2,...,nm}，给出渐近分析。

随机顺序揭示环及其幂中的最后一个孤立顶点：精确枚举与Weibull极限

The Last Isolated Vertex in Random-Order Uncovering of Cycles and Their Powers: Exact Enumeration and Weibull Limits

本文研究随机顺序揭示环及其幂的顶点时最后一个孤立顶点出现的时间，通过逆转过程得到精确枚举公式，并证明普通环的极限为Rayleigh分布，其幂的极限为Weibull分布。

A local injective proof of log-concavity for increasing spanning forests

We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a sin…

On plane permutations

In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $ω$, for which $ω(12\cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It turns out th…

Separation probabilities and analogues of a Zagier-Stanley formula

In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct cycles (or separated) is given by $$ \frac{2 (n-1)! C_m(n+1,k)}{(n+m)(n+1-m)} $$ when $n-k$ is even, where $C_m(n,k)$ is the number of permutations of $n$ elements having $k$ cycles and separating $m$ given elements. As consequences, we obtain the formulas for certain separation probabilities due to Du and Stanley, answering a call of Stanley for simple combinat…

Martingales and descent statistics

This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics inclu…

A central limit theorem for cycles of Mallows permutations

Fix $q\neq 1$, and sample $w\in S_n$ from the Mallows measure. We study the distribution of $C_i(w)$, the number of $i$-cycles, as $n$ grows large. When $q<1$, they are jointly Gaussian, and this more or less follows from known ideas, but the regime $q>1$ behaves quite differently. In particular, we show that the even cycles $C_{2i}(w)$ have a mean and variance of order $n$, and jointly converge to Gaussian random variables, while the odd cycles $C_{2i+1}(w)$ have a bounded mean and variance, and converge to $C_{2i+1}(w^{even})$ or $C_{2i+1}(w^{odd})$ for some explicit random permutations $w^{…

On the Cycle Structure of Mallows Permutations

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $π\in \mathbb{S}_n$ is proportional to $q^{\textrm{inv}(π)}$ where $0<q\le 1$ and $\textrm{inv}(π)$ is the number of inversions in $π$. We show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2}, n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2} \ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-…

马尔可夫链多个同时输出的方差与协方差

Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

本文研究马尔可夫链多个同时输出的方差与协方差，给出方差无界及协方差无界的组合刻画，并说明协方差有界时坐标渐近独立，通过实例验证结果。

Profiles of permutations

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $σ$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $σ$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 \over 2} \log n + O(1)$, and the variance is of the same order. Furthermore, we show that in perm…

按循环类型划分的置换统计量的矩

Moments of permutation statistics by cycle type

该研究引入正则统计量，通过将其分解为部分置换指示函数线性组合再求矩的两步法，推广了置换统计量矩的相关结论，并证明其大数定律与方差性质，同时给出不依赖表示论的自包含证明。

关于随机映射最深环的研究

On the Deepest Cycle of a Random Mapping

本文研究随机映射最大分量中最深环的长度，证明了其经 $\sqrt{n}$ 标准化后的依分布收敛性，并给出期望与方差的极限，同时刻画了最深环顶点占比及最长环顶点不在最大分量中的概率。

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。

Starr猜想的证明以及随机交换置换的对数凹性

Proof of a conjecture by Starr and log-concavity for random commuting permutations

本文证明Starr关于给定联合轨道数的交换置换元组计数渐近猜想，并据此确认典型区间内的对数凹性，同时发展多重Ramanujan和与多极点鞍点渐近方法。

焦磷酸测序中阳性信号的分布

Distributions of positive signals in pyrosequencing

本文推导了焦磷酸测序中阳性信号数量随流动循环次数和核苷酸概率变化的统计分布，并给出均值与方差的显式公式，发现平均阳性信号数近似为流动循环次数的两倍，有助于相关仪器与软件开发。