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.     

Stable multivariate Eulerian polynomials and generalized Stirling permutations

We study Eulerian polynomials as the generating polynomials of the descent statistic over Stirling permutations—a class of restricted multiset permutations. We develop their multivariate refinements by indexing variables by the values at the descent tops, rather than the position where they appear. We prove that the obtained multivariate polynomials are stable, in the sense that they do not vanish whenever all the variables lie in the open upper half-plane. Our multivariate construction generalizes the multivariate Eulerian polynomial for permutations, and extends naturally to r-Stirling and generalized Stirling permutations.The benefit of this refinement is manifold. First of all, the stability of the multivariate generating functions implies that their univariate counterparts, obtained by diagonalization, have only real roots. Second, we obtain simpler recurrences of a general pattern, which allows for essentially a single proof of stability for all the cases, and further proofs of equidistributions among different statistics. Our approach provides a unifying framework of some recent results of Bóna, Brändén, Brenti, Janson, Kuba, and Panholzer. We conclude by posing several interesting open problems.     

Explicit formulas and recurrence relations for higher order Eulerian polynomials

In the paper, the authors present explicit formulas, nonlinear ordinary differential equations, and recurrence relations for Eulerian polynomials, higher order Eulerian polynomials, and their generating functions in terms of the Stirling numbers of the second kind.     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

Alternating Eulerian polynomials and left peak polynomials

We first present grammatical interpretations for the alternating Eulerian polynomials of types A and B. As applications, we then derive several properties of the type B alternating Eulerian polynomials, including recurrence relations, generating function and unimodality. And then, we establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials, which implies that the type B alternating Eulerian polynomials have gamma-vectors that alternate in sign.     

Multivariate stable Eulerian polynomials on segmented permutations

Recently, Nunge studied Eulerian polynomials on segmented permutations, namely generalized Eulerian polynomials, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunge’s conjecture. Our proof is based on Brändén’s stable multivariate Eulerian polynomials. By acting on Brändén’s polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunge’s conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.     

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.     

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén [P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514-531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a (p,q)-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The (p,q)-analogue unifies and generalizes our recent results [H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689-1705] and that of Josuat-Vergès [M. Josuat-Vergés, A q-enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892-1906].     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

Stable multivariate W-Eulerian polynomials

We prove a multivariate strengthening of Brenti?s result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti?s real-rootedness conjecture for Eulerian polynomials of type D, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D, we indicate some methods of attack and pose some related open problems.     

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.     

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.     

Eulerian and even-face ribbon graph minors

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.     

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.     

关于广义Euler多项式和分式的讨论

This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summation formula is given.     

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 some expansions for the Euler Gamma function and the Riemann Zeta function

In the present paper we introduce some expansions which use the falling factorials for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.     

Jucys–Murphy elements,orthogonal matrix integrals,and Jack measures

We study symmetric polynomials whose variables are odd-numbered Jucys–Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.     

Recurrences for Eulerian Polynomials of Type B and Type D

We introduce new recurrences for the type B and type D Eulerian polynomials, and interpret them combinatorially. These recurrences are analogous to a well-known recurrence for the type A Eulerian polynomials. We also discuss their relationship to polynomials introduced by Savage and Visontai in connection to the real-rootedness of the corresponding Eulerian polynomials.