Whitney Numbers Research Articles

A q-analogue of α־-Whitney numbers

We define the (q,??)-Whitney numbers which are reduced to the ??-Whitney numbers when q ? 1. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the ??-Whitney-Lah numbers as a generalization of the r-Whitney-Lah numbers and we introduce their important basic properties.

New family of Whitney numbers

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

A generalization of the $r$-Whitney numbers of the second kind

A generalization of the $r$-Whitney numbers of the second kind

The Generalized <i>r</i>-Whitney Numbers

In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of The relations between Whitney and Stirling numbers are given.

Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

A Note on the Translated Whitney Numbers and Their q-Analogues

This paper presents natural q-analogues for the translated Whitney numbers. Several combinatorial properties which appear to be q-deformations of those classical ones are obtained. Moreover, we give a combinatorial interpretation of the classical translated Whitney numbers of the first and second kind, and their q-analogues in terms of A-tableaux.

The Noncentral Version of the Whitney Numbers: A Comprehensive Study

This paper is a comprehensive study of a certain generalization of Whitney-type and Stirling-type numbers which unifies the classical Whitney numbers, the translated Whitney numbers, the classical Stirling numbers, and the noncentral Stirling (orr-Stirling) numbers. Several identities, applications, and occurrences are also presented.

Context-free grammars for triangular arrays

We consider context-free grammars of the form \(G = \{ f \to f^{b_1 + b_2 + 1} g^{a_1 + a_2 } ,g \to f^{b_1 } g^{a_1 + 1} \} \), where ai and bi are integers subject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn(x) = Σk=0nT(n, k)xk. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn+1(x) = Pn(Tn(x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Branden, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh. Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.

On the Translated Whitney Numbers and their Combinatorial Properties

On the Translated Whitney Numbers and their Combinatorial Properties

The Translated Dowling Polynomials and Numbers.

More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices

In 1973 T.A. Dowling constructed a class of geometric lattices with fixed underlying finite groups. Dowling and M. Benoumhani deduced a number of identities satisfied by the Whitney numbers of these lattices. In addition, Remmel and Wachs gave a partition-theoretical interpretation for these numbers. We continue the study of this interpretation introducing an analogue of Eulerian numbers connected to Whitney numbers of the second kind. Moreover, bijective proofs are given for a number of formulas deduced analytically by Benoumhani.

All 4-dimensional infra-solvmanifolds are boundaries

Infra-solvmanifolds are a certain class of aspherical manifolds which generalize both flat manifolds and almost flat manifolds (i.e., infra-nilmanifolds). Every 4-dimensional infra-solvmanifold is diffeomorphic to a geometric \(4\)-manifold with geometry of solvable Lie type. There were questions about whether or not all 4-dimensional infra-solvmanifolds bound. We answer this affirmatively. On each infra-solvmanifold \(M\) admitting \(\text {Nil}^{3} \times {\mathbb R},\,\text {Nil}^{4},\,\text {Sol}^{3} \times {\mathbb R}\), or \(\text {Sol}_1^{4}\) geometry, an isometric involution with 2-dimensional fixed set is constructed. The Stiefel–Whitney number \(\omega _1^4(M)\) vanishes by a result of R.E. Stong and from this it follows that all Stiefel–Whitney numbers vanish.

Some results on Whitney numbers of Dowling lattices

In this paper, we study some properties of Whitney numbers of Dowling lattices and related polynomials. We answer the following question: there is a relation between Stirling and Eulerian polynomials. Can we find a new relation between Dowling polynomials and other polynomials generalizing Eulerian polynomials? In addition, some congruences for the Dowling numbers are given.

A convolution for complete and elementary symmetric functions

In this paper we give a convolution identity for the complete and elementary symmetric functions. This result can be used to proving and discovering some combinatorial identities involving $r$-Stirling numbers, $r$-Whitney numbers and $q$-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde's convolution identity.

A Generating Function Approach to the Edge Surface Area of the Arrangement Graphs

An important and interesting parameter of an interconnection network is the number of vertices of a specific distance from a specific vertex. This is known as the surface area or the Whitney number of the second kind. It turns out that, in some applications, the number of vertices of a specific distance from a subgraph H is also important. A fundamental starting point is to consider the number of vertices of a specific distance from an edge, which is called the edge surface area. In this paper, we give an explicit formula for the edge surface area of arrangement graphs via the generating function technique. As a direct consequence, it will also provide such explicit formulas for star graphs, alternating group graphs and split stars.

Formula omitted]-Whitney numbers of Dowling lattices

Let G be a finite group of order m≥1. A Dowling lattice Qn(G) is the geometric lattice of rank n over G. In this paper, we define the r-Whitney numbers of the first and second kind over Qn(G), respectively. This concept is a common generalization of the Whitney numbers and the r-Stirling numbers of both kinds. We give their combinatorial interpretations over the Dowling lattice and we obtain various new algebraic identities. In addition, we develop the r-Whitney–Lah numbers and the r-Dowling polynomials associated with the Dowling lattice.

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.

Stiefel–Whitney numbers for singular varieties

AbstractThis paper determines which Stiefel–Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying theF2-vector space of Stiefel–Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces ofRP3bundles. These Stiefel–Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of3bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

Complements and Higher Resonance Varieties of Hyperplane Arrangements

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal.

On the Probability that Certain Compositions Have the Same Number Of Parts

We compute the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts. In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets. Explicit algebraic generating functions and asymptotic probabilities are also computed in the case of pairs of compositions of n into parts at least d, for any fixed natural number d.