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15
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
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Cited by 68 (2 self)
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This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
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Cited by 20 (0 self)
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The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
Effective scalar products of Dfinite symmetric series
 Journal of Combinatorial Theory Series A, 112:1
"... Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We ext ..."
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Cited by 13 (10 self)
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Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to kregular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective DFinite Symmetric Functions”.)
A lower bound on compression of unknown alphabets
 Theoret. Comput. Sci
, 2005
"... Many applications call for universal compression of strings over large, possibly infinite, alphabets. However, it has long been known that the resulting redundancy is infinite even for i.i.d. distributions. It was recently shown that the redudancy of the strings ’ patterns, which abstract the values ..."
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Cited by 10 (3 self)
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Many applications call for universal compression of strings over large, possibly infinite, alphabets. However, it has long been known that the resulting redundancy is infinite even for i.i.d. distributions. It was recently shown that the redudancy of the strings ’ patterns, which abstract the values of the symbols, retaining only their relative precedence, is sublinear in the blocklength n, hence the persymbol redundancy diminishes to zero. In this paper we show that pattern redundancy is at least (1.5 log 2 e) n 1/3 bits. To do so, we construct a generating function whose coefficients lower bound the redundancy, and use Hayman’s saddlepoint approximation technique to determine the coefficients ’ asymptotic behavior. 1
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 8 (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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Cited by 8 (0 self)
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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.
On the problem of uniqueness for the maximum Stirling number(s) of the second kind
 INTEGERS
, 2002
"... Say that an integer n is exceptional if the maximum Stirling number of the second kind S(n, k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x 3/5+ɛ), for any ɛ>0. ..."
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Cited by 7 (0 self)
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Say that an integer n is exceptional if the maximum Stirling number of the second kind S(n, k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x 3/5+ɛ), for any ɛ>0.
The Average Case Analysis Of Algorithms  Saddle Point Asymptotics
 INRIA Rpt
, 1994
"... This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. The series should comprise the following chapters; 1. Symbolic Enumeration and Ordinary Generating Functions; 2. Labelled Struc ..."
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Cited by 7 (1 self)
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This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. The series should comprise the following chapters; 1. Symbolic Enumeration and Ordinary Generating Functions; 2. Labelled Structures and Exponential Generating Functions; 3. Parameters and Multivariate Generating Functions; 4. Complex Asymptotic Methods; 5. Singularity Analysis of Generating Functions; 6. Saddle Point Asymptotics; 7. Mellin Transform Asymptotics; 8. Functional Equations and Generating Functions; 9. Multivariate Asymptotics and Combinatorial Distributions. Chapters 13 have been issued as INRIA Research Report 1888 ("The Average Case Analysis of Algorithms: Counting and Generating Functions", 116 pages, 1993). Chapters 45 as INRIA Research Report 2026 ("The Average Case Analysis of Algorithms: Complex Asymptotics and Generating Functions", 100 pages, 1993). The present report corresponds to Chapter 6 of the series.
Random generation of finitely generated subgroups of a free group
"... We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm ra ..."
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Cited by 5 (3 self)
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We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity O(n) in the RAM model and O(n 2 log 2 n) in the bitcost model.
Universal compression of Markov and related sources over arbitrary alphabets
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2006
"... Recent work has considered encoding a string by separately conveying its symbols and its pattern—the order in which the symbols appear. It was shown that the patterns of i.i.d. strings can be losslessly compressed with diminishing persymbol redundancy. In this paper the pattern redundancy of distri ..."
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Cited by 3 (2 self)
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Recent work has considered encoding a string by separately conveying its symbols and its pattern—the order in which the symbols appear. It was shown that the patterns of i.i.d. strings can be losslessly compressed with diminishing persymbol redundancy. In this paper the pattern redundancy of distributions with memory is considered. Close lower and upper bounds are established on the pattern redundancy of strings generated by Hidden Markov Models with a small number of states, showing in particular that their persymbol pattern redundancy diminishes with increasing string length. The upper bounds are obtained by analyzing the growth rate of the number of multidimensional integer partitions, and the lower bounds, using Hayman’s Theorem.