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19
Consecutive patterns in permutations
 Adv. in Appl. Math
, 2003
"... Abstract. In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of cer ..."
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Cited by 40 (7 self)
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Abstract. In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. Our method is based on the representation of permutations as increasing binary trees and on symbolic methods. 1.
On the Analysis of Linear Probing Hashing
, 1998
"... This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, ..."
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Cited by 26 (8 self)
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This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n3/2), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 2 3, 3.) For sparse tables, the construction cost has expectation O(n), standard deviation O ( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 19 (5 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
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Cited by 18 (7 self)
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We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 16 (6 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Multiple Pattern Avoidance with Respect to Fixed Points and Excedances
, 2004
"... We study the distribution of the statistics `number of fixed points' and `number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statisti ..."
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Cited by 13 (6 self)
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We study the distribution of the statistics `number of fixed points' and `number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statistics. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3.
Fixed points and excedances in restricted permutations
"... In this paper we prove that among the permutations of length n with i fixed points and j excedances, the number of 321avoiding ones equals the number of 132avoiding ones, for all given i, j ≤ n. We use a new technique involving diagonals of nonrational generating functions. This theorem generaliz ..."
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Cited by 13 (7 self)
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In this paper we prove that among the permutations of length n with i fixed points and j excedances, the number of 321avoiding ones equals the number of 132avoiding ones, for all given i, j ≤ n. We use a new technique involving diagonals of nonrational generating functions. This theorem generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. 1.
Dynamical Analysis of αEuclidean Algorithms
, 2002
"... We study a class of Euclidean algorithms related to divisions where the remainder belongs to [α  1, α], for some α 2 [0; 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the socalled "dynamica ..."
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Cited by 9 (3 self)
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We study a class of Euclidean algorithms related to divisions where the remainder belongs to [α  1, α], for some α 2 [0; 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the socalled "dynamical analysis" method, where it is made a deep use of dynamical systems. Here, the dynamical systems of interest have an infinite of branches and they are not markovian, so that the general framework of dynamical analysis is more complex to adapt to this case.
Analytic Variations on Redundancy Rates of Renewal Processes
 IEEE Trans. Information Theory
, 2002
"... Csisz ar and Shields have recently proved that the minimax redundancy for a class of (stationary) renewal processes is ( n) where n is the block length. This interesting result provides a first nontrivial bound on redundancy for a nonparametric family of processes. The present paper gives a precis ..."
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Cited by 8 (5 self)
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Csisz ar and Shields have recently proved that the minimax redundancy for a class of (stationary) renewal processes is ( n) where n is the block length. This interesting result provides a first nontrivial bound on redundancy for a nonparametric family of processes. The present paper gives a precise estimate of the redundancy rate for such (nonstationary) renewal sources, namely, 2 n +O(log n): This asymptotic expansion is derived by complexanalytic methods that include generating function representations, Mellin transforms, singularity analysis and saddle point estimates. This work places itself within the framework of analytic information theory.