Whitney Numbers Research Articles

Zeros distribution and interlacing property for certain polynomial sequences

Abstract In this article, we first prove that the Hankel determinant of order three of the polynomial sequence { P n ( x ) = ∑ k ≥ 0 P ( n , k ) x k } n ≥ 0 {\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P ( n , k ) P\left(n,k) satisfies the recurrence relation P ( n , k ) = ( a 1 n + a 2 ) P ( n − 1 , k ) + ( b 1 n + b 2 ) P ( n − 1 , k − 1 ) , P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P ( n , k ) = 0 P\left(n,k)=0 wherever k ∉ { 0 , 1 , … , n } . k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence ( U n ( x ) ) n ≥ 0 , {\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: U n ( x ) = ( α n x + β n ) U n − 1 ( x ) + ( u n x 2 + v n x ) U n − 1 ′ ( x ) {U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}^{2}+{v}_{n}x){U}_{n-1}^{^{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials.

Probabilistic degenerate Dowling polynomials associated with random variables

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely, the probabilistic degenerate Whitney numbers of the second kind associated with and the probabilistic degenerate Dowling polynomials associated with . We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely, the probabilistic degenerate ‐Whitney numbers of the second kind associated with and the probabilistic degenerate ‐Dowling polynomials associated with , and get similar results to the aforementioned numbers and polynomials.

Valuative invariants for large classes of matroids

AbstractWe study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's ‐polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

A Class of Polynomial Recurrences Resulting in (n/logn,n/log2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {n}}/{\ ext {log}}\\,\\,{\ ext {n}}, {\ ext {n}}/{\ ext {log}}^2{\ ext {n}})$$\\end{document}–Asymptotic Normality

We consider sequences of polynomials that satisfy differential–difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We identify a class of polynomial recurrences that lead to a normal law with the expected value and the variance proportional to n/logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/\\log ~n$$\\end{document} and n/log2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/\\log ^2n$$\\end{document}, respectively. Examples include Stirling numbers of the second kind and other polynomials concerning set partitions as well as polynomials related to Whitney numbers of Dowling lattices.

Alexander and Jones polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice

In this paper, a connection is established between the Jones polynomial of generalized weaving knots of type W(3,n,m) and the Chebyshev polynomial of the first kind. Consequently, it is proved that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices. Additionally, an explicit formula for the coefficients of the Alexander polynomial of weaving knots W(3,n) is introduced, and it is proven that these coefficients satisfy Fox's trapezoidal conjecture.

Some Identities of Fully Degenerate Dowling Polynomials and Numbers

Recently, Kim-Kim introduced the degenerate Whitney numbers of the first and second kind involving the degenerate Dowling polynomials and numbers. In this paper, we introduce the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and derive some identities involving these polynomials. We also obtain the generating functions, expressions, and recurrence relations for the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and other special polynomials and numbers.

The Endomorphism Conjecture for Graded Posets with Whitney Numbers at most 4

We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.

Normal Ordering Associated with $$\lambda$$-Whitney Numbers of the First Kind in $$\lambda$$-Shift Algebra

Normal Ordering Associated with $$\lambda$$-Whitney Numbers of the First Kind in $$\lambda$$-Shift Algebra

Modifications of hyperplane arrangements

This paper is concerned with five kinds of modification of hyperplane arrangements, including elementary lift, parallel translation, coning, one-element extension and restriction to a hyperplane. We show that the combinatorial classification of all hyperplane arrangements of each kind of modification will be characterized by the intersection lattice of the discriminantal or adjoint arrangement. Based on the classifications, a number of combinatorial invariants, including the unsigned coefficients of the Whitney polynomial, Whitney numbers of both kinds, face numbers and region numbers, are constants on those strata associated to the intersection lattice of the discriminantal or adjoint arrangement. Moreover, we further establish the order-preserving relations of those combinatorial invariants and a series of convolution formulae on the characteristic polynomials.

Rank-Metric Lattices

We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.

Asymptotic normality of the Stirling-Whitney-Riordan triangle

Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 + ?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this paper, we show the asymptotic normality of Tn,k and give an asymptotic formula of Tn. As applications, we show the asymptotic normality of many famous combinatorial numbers, such as the Stirling numbers of the second kind, the Whitney numbers of the second kind, the r-Stirling numbers and the r-Whitney numbers of the second kind.

Some Properties of the Parking Function Poset

In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with two well-known polytopes : the associahedron and the permutohedron.

SOME IDENTITIES OF FULLY DEGENERATE DOWLING AND FULLY DEGENERATE BELL POLYNOMIALS ARISING FROM λ-UMBRAL CALCULUS

Recently, Kim–Kim introduced the [Formula: see text]-umbral calculus, in which the [Formula: see text]-Sheffer sequences occupy the central position. In this paper, we introduce the fully degenerate Bell and the fully degenerate Dowling polynomials, and investigate some properties and identities relating to those polynomials with the help of [Formula: see text]-umbral calculus. Here, we note that the fully degenerate Bell poynomials and the fully degenerate Dowling polynomials are, respectively, degenerate versions of the Bell polynomials and the Dowling polynomials, of which the latters are the natural extension of the Whitney numbers of the second kind.

Degenerate Whitney Numbers of First and Second Kind of Dowling Lattices

Dowling constructed Dowling lattice Qn(G), for any finite set with n elements and any finite multiplicative group G of order m, which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kinds for any finite geometric lattice. These numbers for the Dowling lattice Qn(G) are the Whitney numbers of the first kind Vm(n,k) and those of the second kind Wm(n,k), which are given by Stirling number-like relations. In this paper, by `degenerating' such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate r-Whitney numbers of both kinds.

Whitney numbers of matroid extensions and co-extensions

Crapo found that single-element extensions of matroid are equivalent to modular cuts. This paper will study the single-element extension and co-extension of a matroid and establish the upper semi-continuity for the signless coefficients of Whitney polynomials and Whitney numbers on both extensions via its extension lattice, a lattice of all non-empty modular cuts.

Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r-Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r-Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r-Dowling polynomials as a natural extension of the degenerate r-Whitney numbers of the second kind and investigate their properties.

Whitney Numbers of Combinatorial Geometries and Higher-Weight Dowling Lattices

We study the Whitney numbers of the first kind of combinatorial geometries. The first part of the paper is devoted to general results relating the Mobius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices $\mathcal{L}(A)$ of the lattice of subspaces of an $\mathbb{F}_q$-linear space, say $X$, generated by a set of projective points $A \subseteq X$. In this context, we introduce the notion of subspace distribution, and show that partial knowledge of the latter is equivalent to partial knowledge of the Whitney numbers of $\mathcal{L}(A)$. This refines a classical result by Dowling. The most interesting applications of our results are to be seen in the theory of higher-weight Dowling lattices (HWDLs), to which we dovote the second and most substantive part of the paper. These combinatorial geometries were introduced by Dowling in 1971 in connection with fundamental problems in coding theory, and further studied, among others, by Zaslavsky, Bonin, Kung, Brini, and Games. To date, still very little is known about these lattices. In particular, the techniques to compute their Whitney numbers have not been discovered yet. In this paper, we bring forward the theory of HWDLs, computing their Whitney numbers for new infinite families of parameters. Moreover, we show that the second Whitney numbers of HWDLs are polynomials in the underlying field size $q$, whose coefficients are expressions involving the Bernoulli numbers. This reveals a new link between combinatorics, coding theory, and number theory. We also study the asymptotics of the Whitney numbers of HWDLs as the field size grows, giving upper bounds and exact estimates in some cases. In passing, we obtain new results on the density functions of error-correcting codes.

Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

Hankel Transform of the First Form (q,r)-Dowling Numbers

In this paper, the Hankel transform of the generalized q-exponential polynomial of the first form (q, r)-Whitney numbers of the second kind is established using the method of Cigler. Consequently, the Hankel trans- form of the first form (q, r)-Dowling numbers is obtained as special case.

Stieltjes moment properties and continued fractions from combinatorial triangles

Many combinatorial numbers can be placed in the following generalized triangular array [Tn,k]n,k≥0 satisfying the recurrence relation:Tn,k=λ(a0n+a1k+a2)Tn−1,k+(b0n+b1k+b2)Tn−1,k−1+d(da1−b1)λ(n−k+1)Tn−1,k−2 with T0,0=1 and Tn,k=0 unless 0≤k≤n for suitable a0,a1,a2,b0,b1,b2,d and λ. For n≥0, denote by Tn(q) the generating function of the n-th row. In this paper, we develop various criteria for x-Stieltjes moment property and 3-x-log-convexity of Tn(q) based on the Jacobi continued fraction expression of ∑n≥0Tn(q)tn, where x is a set of indeterminates consisting of q and those parameters occurring in the recurrence relation. With the help of a criterion of Wang and Zhu (2016) [36], we show that the corresponding linear transformation of Tn,k preserves Stieltjes moment properties of sequences. Finally, we present some related examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics.