Yaming Yu
Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and w(n) is Lambert's W-function.
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We construct a new parametrization of double sequences $\{A_{n,k}(s)\}_{n,k}$ between $A_{n,k}(0)= \binom{n-1}{k-1}$ and $A_{n,k}(1)= \frac{1}{n!}\stirl{n}{k}$, where $\stirl{n}{k}$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de\,Moivre--Laplace central limit theorem and Goncharov's result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.
Stirling numbers with higher level and records
该研究揭示了更高阶第一类Stirling数与F^α方案中记录数、记录时刻概率分布的关联，并确定了这类Stirling数最大值的位置。
We derive two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind using analytic continuation of a well known identity for the Stirling numbers of the first 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.
For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. Thereby a supplement of the asymptotic analysis for these numbers is established.
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.
New formulas involving Bernoulli and Stirling numbers of both kinds
本文提出了几个涉及两类Bernoulli数和Stirling数的新公式。
For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.
Beta distribution and associated Stirling numbers of the second kind
本文基于贝塔分布的矩，推导出关联第二类斯特林数的公式并给出其上下界。
By considering Eulerian numbers and ordered Stirling numbers of the second and third kinds over a multiset, we generalize identities of Eulerian numbers and Stirling numbers of the second and third kinds and provide $q$-analogs of these generalizations. Using these generalizations, we also compute Eulerian numbers and ordered Stirling numbers of the second kind over a multiset.
Moment Generating Stirling Numbers of the first kind and Applications
本文引入第一类矩生成Stirling数MSN1，作为MSN2的逆元使统计分布的矩可用闭式公式表示，并证明其与r-Stirling数的关系及在任意实数参数下的转换性质。
For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.
Probabilistic degenerate Stirling numbers of the first kind and their applications
本文基于退化累积量生成函数引入第一类概率退化斯特林数，系统研究其性质、恒等式、递推关系与显式表达式，并应用于正态和伽马随机变量。
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.
We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $Π(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the $q…
In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …
Some identities involving $q$-Stirling numbers of the second kind in type B
本文研究B型第二类q-Stirling数，给出联系其与Carlitz主q-Eulerian数的经典恒等式的B型q-类比，并用组合与解析方法证明相关恒等式，同时得到A、B、D型的一些q-恒等式。
It is shown in this note that non-central Stirling numbers s(n,k,a) of the first kind naturally appear in the expansion of derivatives of the product of a power function and a logarithn function. We first obtain a recurrence relation for these numbers, and then, using Leibnitz rule we obtain an explicit formula for these numbers. We also obtain an explicit formula for s(n,1,a), and then derive several combinatorial identities related to these numbers.
In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial $n!$ are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.
Restricted $r$-Stirling numbers of the second kind
研究第二类受限\((r,d)\)-斯特林数，通过施加距离限制推广\(r\)-斯特林数，建立递推关系、公式、生成函数及恒等式等。
In this short note we provide some algebraic identity with a proof exploiting its probabilistic interpretation. We show several consequences of the identity, in particular we obtain a new representation of a Stirling number of second kind, $$ S(n,d)={1\over d!} \sum_{1\leq j_1<j_2<\ldots<j_{d-1}< n} 1\cdot2^{j_{d-1}-j_{d-2}}\cdots d^{j_1}$$ for integers $n\geq d$. Relating this to other known formula for $S(n,d)$ we also obtain $$ \sum_{1\leq j_1\leq j_2\leq \cdots\leq j_{n-d}\leq d} j_1j_2\ldots,j_{n-d} =d! \sum_{1\leq j_1<j_2<\ldots<j_{d-1}< n} 1\cdot2^{j_{d-1}-j_{d-2}}\cdots d^{j_1}.$$ As a…
Sum rules for permutations with fixed points involving Stirling numbers of the first kind
本文提出基于第一类斯特林数的置换固定点求和规则，并推导了二项系数的求和规则，同时探讨了与贝尔数的关系。
Elementary asymptotics for the Stirling numbers of the second kind: The central range
通过初等方法推导第二类斯特林数的局部和中心极限定理，并得到Bell数及分布矩的有效渐近估计，同时基于有限差分方法给出沿多个方向的渐近展开。
The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a λ-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a λ-analogue of the shift algebra (called λ-shift algebra) and investigate normal orderings in the λ-shift algebra. From the normal orderings in the λ-shift algebra, we derive some identities about the λ-analogue of the unsigned Stirl…
In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.
An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind
该研究发现了一个以第二类Stirling数表示Bernoulli数的显式计算公式，为数论相关计算提供了新的显式表达工具。
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both ki…
In the paper, the authors establish an explicit formula for computing Bernoulli polynomials at non-negative integer points in terms of $r$-Stirling numbers of the second kind.
We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of parametric Bernstein-Sato polynomials associated with certain hypergeometric ideals in the Weyl algebra.
We analyze congruence classes of $S(n,k)$, the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for $k\le5$. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for $k=5, 6,..., 20$.
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…
Explicit estimates for the Stirling numbers of the second kind
该研究针对第二类Stirling数$S(n,m)$给出显式渐近紧估计，根据$m$在$\{1,\ldots,n\}$中的区域采用不同概率表示证明结果，推进了组合数的定量估计研究。
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing Euler polynomials and two-parameter Euler polynomials, which are a newly introduced notion, in terms of Stirling numbers of the second kind.
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and $c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\ldots$, $c_k$ cells labeled $k$ and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the \textit{mixed partition numbers}. As an application, we g…
Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c, p^\infty b + d). When a equiv b mod (p-1) or d \le 0, we express them in terms of p-adic binomial coefficients introduced in a recent paper.
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Combinatorial identities related to degenerate Stirling numbers of the second kind
本文研究退化第二类 Stirling 数的性质、恒等式、递推关系与显式表达式，并结合退化 Euler 公式探讨相关多项式和幂级数。
Given an integer g, g > 1, an integer w, -1 < w <g - 1, and a set of g distinct numbers, c_1, ..., c_g, we present a conjectured identity for Stirling numbers of the first kind. We have proven all the equalities in case g < 7; and for the case g = 7, provided w < 4. These expressions arise from an aspect of the study of the dimer-monomer problem on regular graphs.
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.
A formula on Stirling numbers of the second kind $S(n, k)$ is proved. As a corollary, for odd $n$ and even $k$, it is shown that $k!S(n, k)$ is a positive multiple of the greatest common divisor of $j!S(n, j)$ for $k+1\leq j\leq n$. Also, as an application to algebraic topology, some isomorphisms of unstable $K^1$-groups of stunted complex projective spaces are deduced.
Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds
本文研究两类 $B$ 型 poly-Stirling 数的 $q,r$-模拟，建立递推、容斥显式公式与指数生成函数，并推广 Comtet–Lancaster 的广义 Stirling 数三角形等价刻画。
It is known that Bernoulli scheme of independent trials with two outcomes is connected with the binomial coefficients. The aim of this paper is to indicate stochastic processes which are connected with the $q$-polynomial coefficients (in particular, with the $q$-binomial coefficients, or the Gaussian polynomials), Stirling numbers of the first and the second kind, and Euler numbers in a natural way. A probabilistic approach allows us to give very simple proofs of some identities for these coefficients.
In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both kind is given. Namely, Stirling numbers of the first kind appear in the closed formula for the n-th derivative of ln x. In the same way Stirling numbers of the second kind appear in the formula for the n-th derivative of f(e^x), where f(x) is an arbitrary smooth real function. This facts allow us to define Stirling numbers within the frame of differential calc…
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.
Combinatorics of q,r-analogues of Stirling numbers of type B
本文通过引入组合参数，构造了B型两类Stirling数的q-模拟与Broder r-变体，并给出其在正交关系和幂和中的应用。
In the paper, the author establishes some identities which show that the functions $\frac1{(1-e^{\pm t})^k}$ and the derivatives $\bigl(\frac1{e^{\pm t}-1}\bigr)^{(i)}$ can be expressed each other by linear combinations with coefficients involving the combinatorial numbers and the Stirling numbers of the second kind, where $t\ne0$ and $i,k\in\mathbb{N}$.
Partial degenerate Stirling numbers
本文研究退化和不完整第二类斯特林数的组合，采用组合方法并提供渐近结果。
Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials
本文研究Jacobi-Stirling数的二项式展开，证明其下降多项式在特定删除或保留横杠字母数下仅有简单负零点，验证了Gessel-Lin-Zeng猜想的五个无限族，并揭示其顶部重及左半递增性质。
Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind
本文利用 Faà di Bruno 公式、第二类中心阶乘数和第二类 Stirling 数，建立了 sinc 与 sinhc 函数实数幂的级数展开式，并给出相关偏 Bell 多项式闭式公式及组合数恒等式。
On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games
本文研究阶乘与第一类斯特林数的交错和，给出指数生成函数并证明其非负性，将其与递增树博弈、弱序到Tamari格的商及平面树双射相联系，并给出有根树博弈胜负的等价刻画。
$B$-Stirling numbers associated to potential polynomials
本文引入与势多项式相关的第一类和第二类 $B$-Stirling 数，提供递推公式并实现两类数间的相互计算。
Stirling numbers for complex reflection groups
本文将B型Coxeter群的q-模拟Stirling数扩展至所有不可约复反射群G，通过交格的Whitney数定义两类Stirling数，并为G(m,p,n)提供组合解释，同时关联超不变代数的Hilbert级数。
We study mod-$φ$ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when $n$ balls are allocated equiprobably and independently among $θ$ boxes in the regime when $θ$ grows linearly with $n$.
We present various identities in the form of convolutions involving Stirling numbers of both kinds, Lah numbers, and binomial coefficients. Certain convolution polynomials are discussed also. The proofs are based on several series transformation formulas.
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras.
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.
Arithmetic Properties of Mixed Stirling Numbers of the second kind
本文研究第二类混合斯特林数，建立其递推关系与指数生成函数，分析其模素数$p$及$p^2$的性质，推广经典Touchard同余式，揭示其具有区别于经典集合分拆的独特数论特征。
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。
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