Central and local limit theorems applied to asymptotic enumeration. III. Matrix recursions

Bender, Edward A.; Richmond, L. Bruce; Williamson, S. Gill

doi:10.1016/0097-3165(83)90012-2

Cited by 25 publications

(34 citation statements)

References 7 publications

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“…The shape of our results is not unexpected since the central and local limit theorems that we obtain are closely related to matrix recursions developed in an important paper of Bender et al [1983]. The de Bruijn graph is classically associated with the combinatorial construction of de Bruijn sequences, and an early use of it in the context of word enumeration appears in Flajolet et al [1988].…”

Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For supporting

confidence: 64%

“…The finite-state property is then a reflection of the finiteness of all the gaps in the fully constrained case under study. This implies the existence of a matrix representation for our problem, a fact related to the technique of transfer matrices [Bender et al 1983]; (see Section 5.1). Then, Perron-Frobenius properties and their perturbed versions apply, as detailed in Section 5.2; see especially Lemmas 2 and 3.…”

Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For mentioning

confidence: 99%

“…The de Bruijn graph is classically associated with the combinatorial construction of de Bruijn sequences, and an early use of it in the context of word enumeration appears in Flajolet et al [1988]. Bender and Kochman [1993] make an implicit use of this construction combined with the central and local limit theorems of Bender et al [1983] to derive a very general class of estimates for subword counts. This shows the shape of the results that are to be expected in such situations, and our statement in Theorem 3 is definitely along these lines.…”

Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For mentioning

confidence: 99%

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Hidden word statistics

Flajolet

Szpankowski

Vallée

2006

J. ACM

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We consider the sequence comparison problem, also known as "hidden" pattern problem, where one searches for a given subsequence in a text (rather than a string understood as a sequence of consecutive symbols). A characteristic parameter is the number of occurrences of a given pattern w of length m as a subsequence in a random text of length n generated by a memoryless source. Spacings between letters of the pattern may either be constrained or not in order to define valid occurrences. We determine the mean and the variance of the number of occurrences, and establish a Gaussian limit law and large deviations. These results are obtained via combinatorics on words, formal language techniques, and methods of analytic combinatorics based on generating functions. The motivations to study this problem come from an attempt at finding a reliable threshold for intrusion detections, from textual data processing applications, and from molecular biology.

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Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For supporting

confidence: 64%

Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For mentioning

confidence: 99%

Section: λ(U) Is the Largest Eigenvalue Of T (U) (Iii) Finally For mentioning

confidence: 99%

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Hidden word statistics

Flajolet

Szpankowski

Vallée

2006

J. ACM

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“…The work of [BR83] has been greatly expanded upon, but always in a similar framework. For example, it has been extended to matrix recursions [BRW83] and the applicability has been extended from algebraic to algebraico-logarithmic singularities of the form F ∼ (z d − g(x)) q log α (1/(z d − g(x))) [GR92]. The difficult step is always deducing asymptotics from the hypotheses (1.2).…”

Section: Introductionmentioning

confidence: 99%

Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

Pemantle¹,

Wilson²

2008

SIAM Rev.

115

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ABSTRACT:Let {a r : r ∈ N d } be a d-dimensional array of numbers, for which the generating function F (z) := r a r z r is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F .Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations.The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples.

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“…This approach is able to precisely determine the weight enumerators for relatively small lengths, but the computation becomes prohibitively expensive as the truncation lengths increase. Bender et al have shown, in [19], that central and local limit theorems can be derived for the growth of the components of the power of a matrix. This approach would, in principle, allow the Hayman approximation (see [20] for a survey) to be applied to the problem of the weight distribution of convolutional codes.…”

mentioning

confidence: 99%

On the Growth Rate of the Input-Output Weight Distribution of Convolutional Encoders

Ravazzi¹,

Fagnani²

2012

SIAM J. Discrete Math.

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Abstract. In this paper, exact formulae of the input-output weight distribution function and its exponential growth rate are derived for truncated convolutional encoders. In particular, these weight distribution functions are expressed in terms of generating functions of error events associated with a minimal realization of the encoder. Although explicit analytic expressions can be computed for relatively small truncation lengths, the explicit expressions become prohibitively complex to compute as the truncation lengths and the weights increase. Fortunately, a very accurate asymptotic expansion can be derived using the Multidimensional Saddle-Point method (MSP method). This approximation is substantially easier to evaluate and is used to obtain an expression of the asymptotic spectral function, and to prove continuity and concavity in its domain (convex and closed). Finally, this approach is able to guarantee that the sequence of exponential growth rates converges uniformly to the asymptotic limit, and to estimate the speed of this convergence.

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Central and local limit theorems applied to asymptotic enumeration. III. Matrix recursions

Cited by 25 publications

References 7 publications

Hidden word statistics

Hidden word statistics

Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

On the Growth Rate of the Input-Output Weight Distribution of Convolutional Encoders

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