Generating functions for ternary algebras and ternary trees

The main object of study are ternary algebras, i.e., algebras with a trilinear operation. In this class we study finitely generated algebras and their growth, as well as the growth of codimensions of absolutely free algebras and some other varieties. For these purposes we use ordinary generating functions and exponential generating functions (the complexity functions). In the classes of absolutely free, free symmetric, free antisymmetric, and some other algebras we study left nilpotent and completely left nilpotent algebras and varieties. The obtained results are equivalent to the enumeration of ternary trees which contain no forbidden subtrees of a special kind. As the main result, we prove that the complexity functions of the varieties of completely left nilpotent and left nilpotent ternary algebras are algebraic.     

Generating functions for Hurwitz-Hodge integrals

We describe explicit generating functions for a large class of Hurwitz-Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes.Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold Gromov-Witten theory of DM stacks.     

Generating functions for a class ofq-polynomials

Summary Some simple ideas are used here to prove a theorem on generating functions for a certain class of q-polynomials. This general theorem is then applied to derive a fairly large number of known as well as new generating functions for the familiar q-analogues of various polynomial systems including, for example, the classical orthogonal polynomials of Hermite, Jacobi, and Laguerre. A number of other interesting consequences of the theorem are also discussed.     

Generating binary trees with uniform probability

A binary tree is characterized as a sequence of graftings. This sequence is used to construct a Markov chain useful for generating trees with uniform probability. A code for the Markov chain gives a characteristic binary string for the trees. The main result is the calculation of the transition probabilities of the Markov chain. Some applications are pointed out.     

Generating functions and duality for integer programs

We consider the integer program P→max c^′x|Ax=y;xNⁿ . Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λR^m have discrete analogues zC^m, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P^*, the analogue of the dual LP in linear programming.     

Generating well-behaved utility functions for compromise programming

The purpose of this paper is to seek utility functions satisfying a weak condition which guarantees that the utility optimum always belongs to the compromise set. This set is a special subset of the attainable or feasible set, which is generated through the application of the well-known operational research approach called compromise programming. It is shown that there are large families of utility functions satisfying this condition, thus reinforcing the value of compromise programming as a good surrogate of the traditional utility optimum.Thanks are due to the reviewers for their helpful suggestions. The English editing by Ms. Christine Méndez is appreciated. The authors have been supported by the Comisión Interministerial de Ciencia y Tecnología (CICYT), Madrid, Spain.     

Generating functions for certain types of permutations

Generating functions are obtained for certain types of permutations analogous to up-down and down-up permutations. In each case the generating function is a quotient of entire functions; the denominator in each case is φ₀²(x) ? φ₁(x)φ₃(x), where

$\text{φ}{}_{\mathrm{j}}\text{(x)=}\text{∑}\text{n=o}\text{∞}\text{x}{}^{\mathrm{4n+j}}\text{(4n+j)}\text{!.}$

     

Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.     

Generating functions for generalized Hermite polynomials associated with parabolic cylinder functions

ABSTRACT

In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisner? group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using the same procedure. Applications of generating functions obtained by Weisner? group theoretic method are discussed.     

Generating functions for vector partition functions and a basic recurrence relation

ABSTRACT

We define a generalized vector partition function and derive an identity for the generating series of such functions associated with solutions to basic recurrence relations of combinatorial analysis. As a consequence we obtain the generating function of the number of generalized lattice paths and a new version of the Chaundy-Bullard identity for the vector partition function.     

Generating functions for the powers of Fibonacci sequences

Whether or not there is an interaction between two factors in their effects on a dependent variable is often a central question. This paper proposes a general mechanism by which an interaction may arise: (a) the two factors are the same thing — or, at least, have a dimension in common — in the sense that it is meaningful to add (or subtract) them; (b) the sum of them (or the difference between them) is what determines the dependent variable; and (c) the relation between the sum (or difference) and the dependent variable is nonlinear. For example, if several factors contribute to arousal in an additive manner, and the relationship of performance score to arousal is inverted-U, the factors will appear to interact in their joint effect on performance. Psychological, medical, and other scientists are likely to be unfamiliar with the (nonlinear) equations used to express this type of theory. Consequently, the task of promoting and interpreting such ideas will fall to the mathematician and statistician.     

Generating functions via integral transforms

In this paper, we use some integral transforms to derive, for a polynomial sequence {Pn(x)}_n?0, generating functions of the type , starting from a generating function of type , where {γn}_n?0 is a real numbers sequence independent on x and t. That allows us to unify the treatment of a generating function problem for many well-known polynomial sequences in the literature.     

Nontrivial discontinuities of the Golovach functions for trees

The ɛ-search problem on graphs is considered. Properties of the Golovach function, which associates each nonnegative number ɛ with the ɛ-search number, are studied. It is known that the Golovach function is piecewise constant, nonincreasing, and right continuous. Golovach and Petrov proved that the Golovach function for a complete graph on more than five vertices may have nonunit jumps. The jumps of the Golovach function for the case of trees are considered. Examples of trees which disprove the conjecture that the Golovach function has only unit jumps for any planar graph are given. For these examples, the Golovach function is constructed. It is shown that the Golovach function for trees with at most 27 edges has only unit jumps. The same assertion is proved for trees containing at most 28 edges all of whose vertices have degree at most 3. The examples mentioned above have minimum number of edges.     

Generating functions for newton-cotes formulae weights and error terms

From the observation that weights in the Newton-Cotes formulae look like integrals of binomial coefficient polynomials, we are able to evaluate their generating function. We also obtain a generating function for the error terms in the Newton-Cotes formulae by generalizing the Euler-Maclaurin sum formula.     

Generating functions for plane partitions of a given shape

For fixed integers α and β, planar arrays of integers of a given shape, in which the entries decrease at least by α along rows and at least by β along columns, are considered. For various classes of these (α,β)-plane partitions we compute three different kinds of generating functions. By a combinatorial method, determinantal expressions are obtained for these generating functions. In special cases these determinants may be evaluated by a simple determinant lemma. All known results concerning plane partitions of a given shape are included. Thus our approach of a given shape provides a uniform proof method and yields numerous generalizations of known results.     

Entire functions arising from trees

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function f with only two critical values ±1 and no asymptotic values such that f~(-1)([-1, 1]) is ambiently homeomorphic to the given tree. This can be viewed as a generalization of the result of Grothendieck(see Schneps(1994)) to the case of infinite trees. Moreover, a similar idea leads to a new proof of the result of Nevanlinna(1932) and Elfving(1934).