A context-free grammar for the e-positivity of the trivariate second-order Eulerian polynomials

Ma-Ma-Yeh made a beautiful observation that a transformation of the grammar of Dumont instantly leads to the γ-positivity of the Eulerian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the e-positivity of the trivariate second-order Eulerian polynomials first introduced by Dumont in the contexts of ternary trees and Stirling permutations, and independently defined by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.     

On the Numbers of Orbits of Permutations under an Operator Related to Eulerian Numbers

In two previous papers an operator on permutations was introduced and its applications to Eulerian numbers were discussed by means of periods and orbits under the operator. In this paper, observing particular subsequences of permutations, an explicit formula for the number of orbits is given for each period. Several identities concerning the number of orbits and its related numbers are also derived.AMS Subject Classification: 05A05, 05A10.     

Eulerian numbers: A spline perspective

In this paper, the spline interpretations of Eulerian numbers and refined Eulerian numbers are presented. Many classical results about Eulerian numbers can follow from the properties of B-splines directly, and some new results about the refined Eulerian numbers and descent polynomials are also derived. Specifically, the explicit and recurrence formulas for the refined Eulerian numbers and descent polynomials are obtained. This paper also provides a new approach to study Eulerian numbers.     

两个包含欧拉数的恒等式的等价性

基于n维多项式空间中基之间的线性变换，证明了两个包含欧拉数的恒等式是等价．     

两个包含欧拉数的恒等式的等价性(英文)

基于n维多项式空间中基之间的线性变换,证明了两个包含欧拉数的恒等式是等价.     

Asymptotic Eulerian expansions for binomial and negative binomial reciprocals

Asymptotic expansions of any order for expectations of inverses of random variables with positive binomial and negative binomial distributions are obtained in terms of the Eulerian polynomials. The paper extends and improves upon an expansion due to David and Johnson (1956-7).

     

On a generalization of Morisita's model for estimating the habitat preference

The Morisita's model for estimating the habitat preference by the ant lionsGenuroides japonicus is generalized by introducing, in addition to the environmental densitiesa andb, a repulsivity parameter . The probability function of the numberL _n of individuals choosing fine sand to settle when a total ofn ant lions are introduced is examined. A heuristic and the minimum chi-square methods for estimating the parametersa, b and are discussed.     

The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials $F_{m,n,1}\left(x\right)$ and $F_{m,n,2}\left(x\right)$. Then, he defined $A_m(n,k)$ and $B_m(n,k)$ to be the polynomials satisfying $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{A}_m(n,k)x^{n?k}{(x+1)}^k$ and $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{B}_m(n,k)x^{n?k}{(x+1)}^k$. In this paper, we give a combinatorial interpretation of the coefficients of $A_{m+1}(n,k)$ and prove a symmetry of the coefficients, i.e., $\left[m^s\right]A_{m+1}(n,k)=\left[m^{n?s}\right]A_{m+1}(n,n?k)$. We give a combinatorial interpretation of $B_{m+1}(n,k)$ and prove that $B_{m+1}(n,n?1)$ is a polynomial in $m$ with non-negative integer coefficients. We also prove that if $n\ge 6$ then all coefficients of $B_{m+1}(n,n?2)$ except the coefficient of $m^{n?1}$ are non-negative integers. For all $n$, the coefficient of $m^{n?1}$ in $B_{m+1}(n,n?2)$ is $?(n?1)$, and when $n\le 5$ some other coefficients of $B_{m+1}(n,n?2)$ are also negative.     

Eulerian pairs and Eulerian recurrence systems

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating run polynomials for hyperoctahedral groups, flag descent polynomials for hyperoctahedral groups and flag ascent-plateau polynomials for Stirling permutations. We derive some properties of associated polynomials. In particular, we prove the alternatingly increasing property and the interlacing property of the ascent-plateau and left ascent-plateau polynomials for Stirling permutations.     

B样条在一些渐近组合问题中的应用

本文考察了B样条函数及其导数的渐近性质,并给出了收敛阶;考察了经典Eulerian数和两类广义Eulerian数的渐近性质;给出了以Hermite多项式表示的细化Eulerian数的渐近形式.Carlitz等人利用中心极限定理得到Eulerian数渐近公式的逼近阶为43阶.利用样条方法,我们得到更为精确的逼近阶.将样条方法引入到组合数的渐近分析中,为离散对象的研究提供了一种新的分析方法.     

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.     

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversal-concatenation map, we (i) give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, (ii) connect this distribution and the joint distribution (pk, des) over ordinary permutations in terms of generating functions, and (iii) confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.     

Congruences for degenerate number sequences

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article, we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identification we deduce systems of congruences which are analogues and generalizations of the Kummer congruences for the ordinary Bernoulli numbers.     

On the analytic extension of Stirling numbers of the first kind

We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function.     

Relationship between Morisita's model for estimating the environmental density and the generalized Eulerian numbers

The environmental density has been defined (Morisita (1971, Statistical Ecology, the Pennsylvania State University Press, 379–401)) as the value of a habitat expressing its unfavorableness for settling of an individual which has a strong mutual-repulsive influence to other individuals in an environment. Morisita studied mutual repulsive behavior of ant lions (Glenuroides japanicus) and provided a recurrence relation without an explicit solution for the probability distribution of individuals settling in each of two habitats in terms of the environmental densities and the numbers of individuals introduced. In this paper the recurrence relation is explicitly solved; certain interesting properties of the distribution are discussed including its relation to the generalized Eulerian numbers and the estimation of the parameters.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

On weighted q-Daehee polynomials with their applications

In this paper, we first consider a generalization of Kim’s $p$-adic $q$-integral on ${\mathbb{Z}}_p$ including parameters $\alpha$ and $\beta$. By using this integral, we introduce the $q$-Daehee polynomials and numbers with weight $\left(\alpha ,\beta \right)$. Then, we obtain some interesting relationships and identities for these numbers and polynomials. We also derive some correlations among $q$-Daehee polynomials with weight $\left(\alpha ,\beta \right)$, $q$-Bernoulli polynomials with weight $\left(\alpha ,\beta \right)$ and Stirling numbers of second kind.     

The maximal-inversion statistic and pattern-avoiding permutations

We define a new combinatorial statistic, maximal-inversion, on a permutation. We remark that the number M(n,k) of permutations in S_n with k maximal-inversions is the signless Stirling number c(n,n−k) of the first kind. A permutation π in S_n is uniquely determined by its maximal-inversion set . We prove it by making an algorithm for retrieving the permutation from its maximal-inversion set. Also, we remark on how the algorithm can be used directly to determine whether a given set is the maximal-inversion set of a permutation. As an application of the algorithm, we characterize the maximal-inversion set for pattern-avoiding permutations. Then we give some enumerative results concerning permutations with forbidden patterns.     

Compatible Eulerian circuits in Eulerian (di)graphs with generalized transition systems

A transition in a graph is defined as a pair of adjacent edges. A transition system of an Eulerian graph refers to a set of partitions such that for each vertex of the graph, there corresponds to a partition of the set of edges incident to the vertex into transitions. A generalized transition system $F\left(G\right)$ over a graph $G$ defines a set of transitions over $G$. A compatible Eulerian circuit of an Eulerian graph $G$ with a generalized transition system $F\left(G\right)$ is defined as an Eulerian circuit in which no two consecutive edges form a transition defined by $F\left(G\right)$. In this paper, we further introduce the concept of weakly generalized transition system which is an extension of the generalized transition system and prove some Ore-type sufficient conditions for the existence of compatible Eulerian circuits in Eulerian graphs with (weakly) generalized transition systems and obtain corresponding results for Eulerian digraphs. Our conditions improve some previous results due to Jackson and Isaak, respectively.