The generalized Fibonomial matrix

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k+1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples.     

A finite difference approach to degenerate Bernoulli and Stirling polynomials

Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Calculation of particular values of the polynomials yields some binomial identities. Properties of the polynomials are established and several factorization results are proven and conjectured.     

The Combinatorics of Polynomial Sequences

We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of U-graph and give examples of binomial prefabs of U-graphs. The umbral composition of U-graphs provides an interpretation of umbral composition of polynomial sequences in Sees. 3 and 5. Rota's interpretation of the Stirling numbers of the first kind as sums of the Mobius function in the partition lattice inspired our model for inverse sequences of binomial type in Sec. 4. Section 6 contains combinatorial proofs of several operator-theoretic results. The actions of shift operators and delta operators are explained in set-theoretic terms. Finally, in Sec. 6 we give a model for cross sequences and Sheffer sequences which is consistent with their decomposition into sequences of binomial type. This provides an interpretation of shift-invariant operators. Of course, all of these interpretations require that the coefficients involved be integer and usually non-negative as well.     

Operatorenkalkül über freien Monoiden I: Strukturen

In part I algebraic structures (esp. rings) on the sets of polynomials and formal power series on an at most countable alphabetA are considered. Given a partial order onA the words ofA ^* are mixed together in consistence with it. It is shown that the structures derived are associative iff the given partial order is of linear type. The coefficients appearing at these operations are identified as generalizations of the ordinary binomial coefficients and a number of relations involving them are listed up.(Part II will bring a generalization ofRota's theory of polynomial sequences of binomial type to the structures studied in I.In Part III the theory of special binomial systems will be continued until the analogue of Lagrange inversion and a short development of generalized Sheffer polynomials will be given).     

Divided Differences and Combinatorial Identities

We present an algebraic theory of divided differences which includes confluent differences, interpolation formulas, Liebniz's rule, the chain rule, and Lagrange inversion. Our approach uses only basic linear algebra. We also show that the general results about divided differences yield interesting combinatorial identities when we consider some suitable particular cases. For example, the chain rule gives us generalizations of the identity used by Good in his famous proof of Dyson's conjecture. We also obtain identities involving binomial coefficients, Stirling numbers, Gaussian coefficients, and harmonic numbers.     

Fibonacci polynomials: compositions and cyclic products

Fibonacci (alias Chebyshev) polynomials enjoy particular composition properties. These can be seen (and proved) from a combinatorial perspective by interpreting these polynomials as matching polynomials. An enumerative technique for cyclic structures is applied to obtain a generating polynomial identity for cyclic products of binomial coefficients in terms of matching polynomials.     

基于一个抽球模型的组合恒等式及组合解释

本文研究了抽球概率模型的问题.利用概率方法,获得了关于第一类Stirling数和广义可重复二项式系数的无限求和形式的组合恒等式以及有关组合解释,推广了Stirling数和二项式系数的无限求和结果.     

Operatorenkalkül über freien Monoiden II: Binomialsysteme

In this part starting from a generalization of the binomial theorem a development of Rota's theory of polynomial sequences of binomial type to the case of countably many noncommuting variables is given. Translation invariance of operators gives the relation to the formal power series considered inI. For a special class of binomial systems there are given a number of characterizations, such as a generalized Rodrigues formula. In the case of an analogue of the Newton polynomials those are used for a study of generalized Stirling numbers. (In III the theory of binomial systems of diagonal type will be continued until an analogue of Lagrange inversion and a short development of the theory of generalized Sheffer polynomials will be given.)     

Singular Polynomials of Generalized Kasteleyn Matrices

Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK* (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.     

Lattice Paths and Positive Trigonometric Sums

A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coefficients is given. The proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer coefficients. Some special cases of the q -analogue conjectured by Bressoud are established, and new conjectures are given. January 22, 1997. Date revised: July 9, 1997.     

Some combinatorial conjectures for shifted Jack polynomials

We present several combinatorial conjectures related to Jack generalized binomial coefficients, or equivalently to shifted Jack polynomials. We prove these conjectures when the degree of these polynomials is 5.     

An algebraic approach to assignment problems

For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.     

The degree of convergence of entire functions and generalized type

Summary The generalized order and generalized type of an entire function have been considered in this paper, using arbitrary growth functions. In place of the usual Taylor series expansion, polynomial series expansion having polynomial coefficients of an entire function have been considered and formula for generalized type obtained in terms of the polynomial coefficients. In the end, a result characterizing the set of entire functions of positive generalized order and finite type in terms of their degree of convergence on general sets has been obtained.     

The multi-indexed partitional

In this paper we present the multi-indexed partitional. The structure is a generalization of the partitional of Foata and Schützenberger, and gives a combinatorial interpretation for certain multivariate polynomial sequences of binomial type.     

The characteristic polynomial of a multiarrangement

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

Monogenic pseudo‐complex power functions and their applications

The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd.     

二项式系数倒数级数恒等式

利用已知级数,通过裂项构造出一批新的二项式系数倒数级数,它们的分母分别含有1到4个奇因子与二项式系数的乘积表达式.所给出二项式系数倒数级数的和式是封闭形的.     

Cubic Pisot unit combinatorial games

Generalized Tribonacci morphisms are defined on a three letters alphabet and generate the so-called generalized Tribonacci words. We present a family of combinatorial removal games on three piles of tokens whose set of P{\mathcal{P}} -positions is coded exactly by these generalized Tribonacci words. To obtain this result, we study combinatorial properties of these words like gaps between consecutive identical letters or recursive definitions of these words. β-Numeration systems are then used to show that these games are tractable, i.e., deciding whether a position is a P{\mathcal{P}} -position can be checked in polynomial time.     

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.