Dmitry V. Kruchinin, Vladimir V. Kruchinin
In the paper, 2 explicit formulas for the Euler numbers of the second kind are obtained. Based on those formulas a exponential generating function is deduced. Using the generating function some well-known and new identities for the Euler number of the second kind are obtained.
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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.
Identities of Eulerain and ordered Stirling numbers over a multiset
通过考虑多重集上的欧拉数和第二、第三类有序Stirling数，我们推广了欧拉数和第二、第三类有序Stirling数的恒等式，并给出了这些推广的$q$ -类似物.利用这些推广，我们还计算欧拉数和第二类有序斯特林数的多集。
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijection…
Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind
本文初步统一和推广了涉及函数$\frac{\pm1}{e^{\pm t}-1}$及其导数的八个恒等式。利用其中的一个恒等式，作者建立了计算欧拉多项式和双参数欧拉多项式的两个显式公式。
Colorful Proofs of the Generating Formulas for Signed and Unsigned Stirling Numbers of the First Kind
我们描述的标准生成公式的无符号和有符号的斯特林数的第一类，从一个自然的组合解释的基础上循环着色排列的证明。
Euler-Mahonian Statistics On Ordered Set Partitions (II)
本文研究了生成函数与第二类$p,q$ -Stirling数有关的有序集分拆的统计问题。本文的主要目的是提供\stein（Arxiv：math.CO/0605670）的所有结构的双射证明。我们的基本思想是通过一种路径图对有序划分进行编码，并探索后者结构的丰富组合性质。我们还给出了关于词的统计反演数和主指数等分布的MacMahon定理的一个剖分形式。
The generating function of planar Eulerian orientations
该研究通过将生成函数表示为简单超几何级数的反函数，解决了平面欧拉定向及4价平面欧拉定向的计数问题，验证了此前的渐近行为预测，并发现了4价情形下的意外组合关联。
Asymptotics of the Euler transform of Fibonacci numbers
OEIS（斐波那契数的欧拉变换）中序列A166861的生成函数是Product_{k>0} 1/（1-x^k）^F（k），其中F（k）是斐波那契数。本文分析了更一般的母函数U（x）= Product_{k>0} 1/（1-x^k）^F（k+z），其中z是非负整数，它为OEIS中的序列A166861（z=0），A200544（z=1）和A260787（z=2）提供了渐近性.
Strong q-log-convexity of the Eulerian polynomials of Coxeter groups
本文利用Coxeter群的指数母函数证明了Coxeter群的欧拉多项式的强$q$ -对数凸性。本文的证明是基于指数Riordan arraya理论和一个判定多项式序列强$q$ -对数凸性的准则，其生成函数可以由连分数给出.作为推论，我们统一地得到了$A_n,B_n$型欧拉多项式、其$q$ -类似多项式以及与等差数列$\{a,a+d,a+2d,a+3d,\ldots\}$相关联的广义欧拉多项式的强$q$ -对数凸性.
Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain $q$-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of t…
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。
A Dirichlet Generating Function for the Coefficients of Euler's Pentagonal Number Theorem
本文通过布罗维奇型积分方法，为欧拉五边形数定理系数建立了狄利克雷生成函数的积分表示，并推导了其渐近行为及显式公式。
We observe that three context-free grammars of Dumont can be brought to a common ground, via the idea of transformations of grammars, proposed by Ma-Ma-Yeh. Then we develop a unified perspective to investigate several combinatorial objects in connection with the bivariate Eulerian polynomials. We call this approach the Dumont ansatz. As applications, we provide grammatical treatments, in the spirit of the symbolic method, of relations on the Springer numbers, the Euler numbers, the three kinds of peak polynomials, an identity of Petersen, and the two kinds of derivative polynomials, introduced…
The Tchebyshev transforms of the first and second kind
我们给出了深入的研究的第一和第二种偏序集的切比雪夫变换，最近发现的Hetyei。切比雪夫变换（第一类）保留了理想的组合性质，包括欧拉性（由于Hetyei）和EL-可壳性。它也是对标志向量的线性变换。当限制为欧拉偏序集时，它对应于定向拟阵的Billera，Escherborg和Readdy Ω映射.一个结果是cd指数的非负性得以保持。第二类切比雪夫变换是拟对称函数空间QSym上的霍普夫代数自同态。它与Stembridge的Eulerian偏序集的峰值计数器一致，但与一般偏序集不同。确定了完整的光谱，推广了Billera，Hsiao和van Willigenburg的工作。引入偏序集的B型拟对称函数。像Escherborg的偏序集的经典拟对称函数一样，这个映射是关于拟对称函数QSym的余同态。欧米茄地图之间的相似之处，埃克斯堡的r-签署伯克霍夫变换，和切比雪夫变换激励一般的研究链地图。一个这样的事件，第二类链映射，是一个霍普夫代数自同态的准对称函数QSym和一个实例阿吉亚尔，Bergeron和Sottile的结果对终对象的范畴内的组合霍普夫代数。与此相反，第一类链映射既是B型拟对称函数BQSym上的代数映射，又是其余自同态。
In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely, \begin{eqnarray*} A_n(p,q)=\sum_{π\in \mathfrak{S}_n}p^{{\rm odes}(π)}q^{{\rm edes}(π)} \end{eqnarray*} for $n\geq 1$, where ${odes}(π)$ and ${edes}(π)$ enumerate the number of descents of permutation $π$ in odd and even positions, respectively. In this paper, we build an exponential generating function for $A_{n}(p,q)$ and establish an explicit formula for $A_{n}(p,q)$ in terms of Eulerian polynomials $A_{n}(q)$ and $C(q)$, the generating function for Catalan numbers. In certain special case, we set up a connection …
Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups
为了得到欧拉数的指数母函数公式的q-模拟，Shareshian和Wachs引入了欧拉拟对称函数。它们是通过对称群定义的，应用稳定和非稳定的主要专业化产生置换统计量的联合分布的公式。我们考虑循环群与对称群的圈积，也称为着色置换群。我们利用这个群来引入有色欧拉拟对称函数，它是欧拉拟对称函数的推广。我们推导出这些有色欧拉拟对称函数的生成函数的公式，这减少到一个公式的Shareshian和Wachs的欧拉拟对称函数。我们表明，应用稳定和不稳定的主要专业化产生的联合分布的有色排列统计量，推广的Shareshian-Wachs q模拟欧拉公式，公式的Foata和韩，和一个公式的周和Gessel的公式。
Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coeffi…
A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.
The $m$th-order Eulerian Numbers
本文定义了 $m$ 阶欧拉数及其组合解释，研究了递推关系、行生成函数、行和、欧拉分数性质，并给出与第二类斯特林数的反演关系及精确值表达式。
We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic examples of our new results are presented in the article to motivate the formulations of the generalized factorization theorems. Applications of these new factorization results include new identities involving the Euler partition function and the generalized sum-of-divisors functions, the Möbius function, Euler's totient function, the Liouville lambda function, vo…
Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences.
The dimensions and Euler characteristics of M. Kontsevich's graph complexes
我们提供了M的（分次）维数的生成函数。普通图的Kontsevich图复形。该母函数可用于计算各回路阶的欧拉示性数。此外，我们表明，图与重边可以省略这些图复合。
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions, including for binary and n-ary partitions.
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 $G_{m,n}$.
A bi-Stirling-Euler-Mahonian polynomial
该研究受（重）混合欧拉数相关研究启发，给出Nadeau和Tewari提出的重混合欧拉数一个子族的组合解释，证明其可表示为排列对应四类统计量的生成多项式，推广了两类已知多项式。
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers
给出了凸（sn+2）-边形沿不相交对角线划分为（sj+2）-边形的次数公式的一个简短证明，其中1 ≤ j ≤ n-1.换句话说，我们考虑将一个（sn+2）-边形剖分为若干块，这些块可以进一步细分为（s+2）-边形。该公式推广了经典的多边形剖分数公式：Euler-Catalan数、Fuss数和Kirkman-Cayley数。我们的证明是初等的，没有使用生成函数的方法。
We give a short proof, using generating functions, for a polynomial congruence for Eulerian polynomials first proved, using arrangements of hyperplanes, by Yoshinaga and later proved, using roots of unity, by Iijima, Sasaki, Takahashi, and Yoshinaga.
Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array $[\mathcal {T}_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} \mathcal {T}_{n,k}=λ(a_0n+a_1k+a_2)\mathcal {T}_{n-1,k}+(b_0n+b_1k+b_2)\mathcal {T}_{n-1,k-1}+\frac{cd}λ(n-k+1)\mathcal {T}_{n-1,k-2} \end{equation*} with $\mathcal {T}_{0,0}=1$ and $\mathcal {T}_{n,k}=0$ unless $0\le k\le n$. We derive a functional transformation for its row-generating function $\mathcal{T}_n(x)$ from the…
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。
We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.
In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating fu…
In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $γ_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $π^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's cons…
A probabilistic approach to $q$-polynomial coefficients, Euler and Stirling numbers
已知双结局独立试验的Bernoulli方案与二项式系数有关。本文的目的是指出随机过程与$q$ -多项式系数（特别是与$q$ -二项式系数，或高斯多项式），第一和第二类Stirling数，和欧拉数以自然的方式。一个概率的方法使我们能够给这些系数的一些身份非常简单的证明。
On some expansions for the Euler Gamma function and the Riemann Zeta function
本文介绍了Euler Γ函数和黎曼ζ函数的几种基于下降因子的展开式。在证明中，我们使用了法迪布鲁诺公式，贝尔多项式，潜在的多项式，Mittag-Leffler多项式，导数多项式和特殊的数字（欧拉数和斯特林数的两种）。我们研究了该级数的收敛速率并给出了一些数值例子。
Eulerian triangulations: two-point function and hull perimeter statistics
该研究在切片生成函数的新递推关系框架下，推导了平面欧拉三角剖分的距离相关两点函数，得到更精细的含外壳周长控制的生成函数，还给出了无限大随机平面欧拉三角剖分外壳周长统计的显式公式，推广了已有相关结果。
Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups
得到了对称群中交换m元组个数的生成函数。定义了作用在流形X上的有限群G的“轨道欧拉示性数”的自然序列.我们的定义推广了X/G的普通欧拉示性数和弦论轨道折叠欧拉示性数。我们的公式交换m元组的基础公式，推广的结果麦克唐纳和Hirzebruch-Hyndham关于普通和弦理论的欧拉示性数的对称产品。
We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.
A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind
我们表明，只有正交多项式的生成函数的形式$F(x z - αz^2)$是超球面，厄米，和切比雪夫多项式的第一类。对于特殊的$F$，这是这种情况下，我们然后完成了更一般的生成函数$F(x w(z) - R(z))$的正交多项式的分类。
Euler characteristics of affine ADE Nakajima quiver varieties via collapsing fibers
本文通过研究GIT映射变分中的坍缩纤维，证明了比较仿射ADE中岛箭图簇欧拉示性数生成级数的普适替换公式，统一并推广了已有结果，并应用于Kleinian轨形非交换Quot概形及A型李代数表示论。
Valuations in Nilpotent Minimum Logic
欧拉特征线可以定义为有限分配格上的一种特殊赋值。这项工作开始于一些简要考虑的作用，欧拉特征的NM代数，代数对应的幂零最小逻辑。然后，我们引入了一个新的估值，一个修改后的版本的欧拉特征，我们称之为幂等欧拉特征。我们表明，新的估值编码NM命题逻辑的公式信息。
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