Matrices related to the Bell polynomials

In this paper, we study the matrices related to the partial exponential Bell polynomials and those related to the Bell polynomials with respect to Ω. As a result, the factorizations of these matrices are obtained, which give unified approaches to the factorizations of many lower triangular matrices. Moreover, some combinatorial identities are also derived from the corresponding matrix representations.     

Identities for partial Bell polynomials derived from identities for weighted integer compositions

We discuss closed-form formulas for the (n, k)th partial Bell polynomials derived in Cvijovi? (Appl Math Lett 24:1544–1547, 2011). We show that partial Bell polynomials are special cases of weighted integer compositions, and demonstrate how the identities for partial Bell polynomials easily follow from more general identities for weighted integer compositions. We also provide short and elegant probabilistic proofs of the latter, in terms of sums of discrete integer-valued random variables. Finally, we outline further identities for the partial Bell polynomials.     

Paradeterminants and partition polynomials

We study Bell polynomials by using functions of triangular matrices (parapermanents and paradeterminants). Some combinatorial identities and relationships between these functions and the Stirling numbers of the first and second kinds are established. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1457–1469, November, 2008.     

A compehensive study of $${\vavec{}}$$-Dowling polynomials

After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones.     

On Touchard polynomials

J. Touchard in his work on the cycles of permutations generalized the Bell polynomials in order to study some problems of enumeration of the permutations when the cycles possess certain properties.In the present paper (considering Touchards's generalization) we introduce and study a class of related polynomials. An exponential generating function, recurrence relations and connections with other well-known polynomials are obtained. In special cases, relations with Stirling number of the first and second kind, as well as with other numbers recently studied are derived. Finally, a combinatorial interpretation is discussed.     

Identities on Bell polynomials and Sheffer sequences

In this paper, we study exponential partial Bell polynomials and Sheffer sequences. Two new characterizations of Sheffer sequences are presented, which indicate the relations between Sheffer sequences and Riordan arrays. Several general identities involving Bell polynomials and Sheffer sequences are established, which reduce to some elegant identities for associated sequences and cross sequences.     

Covariance of Lax Pairs and Integrability of the Compatibility Condition

We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DT). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DT via a version of non-Abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov–Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DT symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville–von Neumann equations for the density matrix are considered as illustrations.     

Bell polynomials and binomial type sequences

This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples.     

Some identities of Bell polynomials

We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.     

On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations

A direct and unifying scheme for the disclosure of bilinear Bäcklund transformations and linear Lax systems associated with soliton equations is presented. The scheme is based on a concept of scale invariance and on the use of a class of partitional polynomials: the binary Bell polynomials. The applicability of the procedure is tested on a variety of soliton equations.     

Special matchings and Kazhdan-Lusztig polynomials

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W, and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials.     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

r-Bell polynomials in combinatorial Hopf algebras

We introduce partial r-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.     

Green polynomials at roots of unity and Springer modules for the symmetric groups

We consider the Green polynomials at roots of unity. We obtain a recursive formula for the Green polynomials at roots of unity whose orders do not exceed some positive integer. The formula is described in a combinatorial manner. The coefficients of the recursive formula are realized by the cardinality of a set of permutations. The formula gives an interpretation of a combinatorial property on a family of graded modules for the symmetric group in terms of representation theory.     

Two Kinds of Numbers and Their Applications

C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.     

Identities among certain triangular matrices

The object of this paper is to develop the ideas introduced in the author's paper [1] on matrices which generate families of polynomials and associated infinite series. A family of infinite one-subdiagonal non-commuting matrices Q_m is defined, and a number of identities among its members are given. The matrix Q₁ is applied to solve a problem concerning the derivative of a family of polynomials, and it is shown that the solution is remarkably similar to a conventional solution employing a scalar generating function. Two sets of infinite triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al. [24-26].     

有关调和数的更多渐近展开公式

基于指数型完全Bell多项式,建立了一个一般调和数渐近展开式,并给出展开式中系数的相应递推关系.由生成函数方法进一步推导出这些系数的具体表达式.另外,我们建立了两个在对数项里只含有奇数或偶数次幂项的lacunary调和数渐近展开式,     

一个(3+1)维非线性方程的B(a)cklund变换和精确解

利用双Bell多项式方法构造了一个(3+1)维非线性方程的双线性形式,得到了该方程的双线性Bcklund变换和相应的Lax对.同时利用Riemann theta函数,获得了该方程的周期波解.     

Combinatorics of the K-theory of affine Grassmannians

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions – Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.