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12
On Asymptotics Of Certain Recurrences Arising In Universal Coding
 Problems of Information Transmission
, 1997
"... Ramanujan's Qfunction and the so called "tree function" T (z) defined implicitly by the equation T (z) = ze T (z) found applications in hashing, the birthday paradox problem, random mappings, caching, memory conflicts, and so forth. Recently, several novel applications of these fun ..."
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Ramanujan's Qfunction and the so called "tree function" T (z) defined implicitly by the equation T (z) = ze T (z) found applications in hashing, the birthday paradox problem, random mappings, caching, memory conflicts, and so forth. Recently, several novel applications of these functions to information theory problems such as linear coding and universal portfolios were brought to light. In this paper, we study them in the context of another information theory problem, namely: universal coding which was recently investigated by Shtarkov et al. [Prob. Inf. Trans., 31, 1995]. We provide asymptotic expansions of certain recurrences studied there which describe the optimal redundancy of universal codes. Our methodology falls under the so called analytical information theory that was recently applied successfully to a variety of information theory problems. Key Words: Source coding, multialphabet universal coding, redundancy, minimum description length, analytical information theory, si...
Computer Algebra Libraries for Combinatorial Structures
, 1995
"... This paper introduces the framework of decomposable combinatorial structures and their traversal algorithms. A combinatorial type is decomposable if it admits a specification in terms of unions, products, sequences, sets, and cycles, either in the labelled or in the unlabelled context. Many properti ..."
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Cited by 12 (0 self)
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This paper introduces the framework of decomposable combinatorial structures and their traversal algorithms. A combinatorial type is decomposable if it admits a specification in terms of unions, products, sequences, sets, and cycles, either in the labelled or in the unlabelled context. Many properties of decomposable structures are decidable. Generating function equations, counting sequences, and random generation algorithms can be compiled from specifications. Asymptotic properties can be determined automatically for a reasonably large subclass. Maple libraries that implement such decision procedures are briefly surveyed (LUO, combstruct, equivalent). In addition, libraries for manipulating holonomic sequences and functions are presented (gfun, Mgfun).
Symbolic Asymptotics: Multiseries of Inverse Functions
, 1997
"... We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given explog function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddlepoint method is then treated in detail. ..."
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We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given explog function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddlepoint method is then treated in detail.
Partially directed paths in a wedge
 Journal of Combinatorial Theory, Series A
"... The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a mo ..."
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The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y = ±pX, and an asymmetric wedge defined by the lines Y = pX and Y = 0, where p> 0 is an integer. We prove that the growth constant for all these models is equal to 1+ √ 2, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when p = 1. From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1. The functional recurrences are solved by a variation of the kernel method, which we call the “iterated kernel method”. This method appears to be similar to the obstinate kernel method used by BousquetMélou (see, for example, references [5, 6]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θfunctions, and have natural boundaries on the unit circle.
The number of hierarchical orderings
 Order 21 (2004) 83—89; arXiv:math/0307064; MR2128036 (2005k:06012
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A Mathematica Package for Computing Asymptotic . . .
"... We describe a simple package for computing a fundamental system of certain formal series solutions, up to a prescribed order, of a given Pfinite recurrence equation. These solutions can be viewed as describing the asymptotic behavior of sequences satisfying the recurrence. ..."
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Cited by 4 (1 self)
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We describe a simple package for computing a fundamental system of certain formal series solutions, up to a prescribed order, of a given Pfinite recurrence equation. These solutions can be viewed as describing the asymptotic behavior of sequences satisfying the recurrence.
Equations in finite semigroups: explicit enumeration and asymptotics of solution numbers
 J. COMBIN. THEORY SER. A
, 2004
"... We study the number of solutions of the general semigroup equation in one variable, X α = X β, as well as of the system of equations X 2 = X, Y 2 = Y, XY = Y X in H ≀ Tn, the wreath product of an arbitrary finite group H with the full transformation semigroup Tn on n letters. For these solution nu ..."
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Cited by 3 (1 self)
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We study the number of solutions of the general semigroup equation in one variable, X α = X β, as well as of the system of equations X 2 = X, Y 2 = Y, XY = Y X in H ≀ Tn, the wreath product of an arbitrary finite group H with the full transformation semigroup Tn on n letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concerning the first mentioned problem generalize earlier results by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967), 122–135) on the number of idempotents in Tn, and a partial result of Dress and the second author (Adv. in Math. 129 (1997), 188–221). Among the asymptotic tools employed are Hayman’s method for the estimation of coefficients of analytic functions and the Poisson summation formula.
Extended Admissible Functions And Gaussian Limiting Distributions
 Mathematics of Computation
"... We consider an extension of Hayman's notion of admissibility to bivariate generating functions f(z; u) that have the property that the coecients a nk satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a big class of funct ..."
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We consider an extension of Hayman's notion of admissibility to bivariate generating functions f(z; u) that have the property that the coecients a nk satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a big class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is presented, too. We apply this concept to various combinatorial examples. 1.
HAYMAN ADMISSIBLE FUNCTIONS IN SEVERAL VARIABLES
"... Abstract. An alternative generalisation of Hayman’s admissible functions ([17]) to functions in several variables is developed and a multivariate asymptotic expansion for the coefficients is proved. In contrast to existing generalisations of Hayman admissibility ([7]), most of the closure properties ..."
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Abstract. An alternative generalisation of Hayman’s admissible functions ([17]) to functions in several variables is developed and a multivariate asymptotic expansion for the coefficients is proved. In contrast to existing generalisations of Hayman admissibility ([7]), most of the closure properties which are satisfied by Hayman’s admissible functions can be shown to hold for this class of functions as well. 1.
Computer Science Journal of Moldova, vol.13, no.1(37), 2005 An Automatic Proof of Euler’s Formula
"... In this information age, everything is digitalized. The encoding of functions and the automatic proof of functions are important. This paper will discuss the automatic calculation for Taylor expansion coefficients, as an example, it can be applied to prove Euler’s formula automatically. ..."
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In this information age, everything is digitalized. The encoding of functions and the automatic proof of functions are important. This paper will discuss the automatic calculation for Taylor expansion coefficients, as an example, it can be applied to prove Euler’s formula automatically.