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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
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 108 (3 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.
UNLABELED (2 + 2)FREE POSETS, ASCENT SEQUENCES AND PATTERN AVOIDING PERMUTATIONS
"... Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of ..."
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Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of D8, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2 + 2)free posets, chord diagrams and permutations. Our bijections preserve numerous statistics. We also determine the generating function of these classes of objects, thus recovering a series obtained by Zagier for chord diagrams. That this series also counts (2 + 2)free posets seems to be new. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3¯152¯4, and enumerate those permutations, thus settling a conjecture of Lara Pudwell. 1.
Random sampling of sparse trigonometric polynomials
 Appl. Comput. Harm. Anal
, 2006
"... We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, ..."
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Cited by 70 (22 self)
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We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
Recounting the rationals
 Amer. Math. Monthly
"... prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtai ..."
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Cited by 52 (3 self)
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prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
Faces of generalized permutohedra
"... Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explici ..."
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Cited by 50 (3 self)
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Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for hvectors and γvectors involving descent statistics. This includes a combinatorial interpretation for γvectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on nonnegativity of γvectors. We calculate explicit generating functions and formulae for hpolynomials of various families of graphassociahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb’s problem. We give (and conjecture) upper and lower bounds for f, h, and γvectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.
A sequential importance sampling algorithm for generating random graphs with prescribed degrees
, 2006
"... Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a ra ..."
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Cited by 43 (1 self)
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Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random
Generalized cluster complexes and Coxeter combinatorics
 Int. Math. Res. Notices
"... and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial) ..."
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Cited by 41 (1 self)
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and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial)
Asymmetric multiple description lattice vector quantizers
 IEEE Trans. Inf. Theory
, 2002
"... Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, a ..."
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Cited by 34 (3 self)
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Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The highrate asymptotic performance of the quantizer is also studied. We evaluate the ratedistortion performance of the quantizer and compare it to known informationtheoretic bounds. The highrate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, highrate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.
On the existence of similar sublattices
 Canad. J. Math
, 1999
"... Partial answers are given to two questions. When does a lattice Λ contain a sublattice Λ ′ of index N that is geometrically similar to Λ? When is the sublattice “clean”, in the sense A similarity σ of norm c is a linear map from Rn to Rn such that σu · σv = c u · v for u,v ∈ Rn. Let Λ be an ndimens ..."
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Cited by 33 (7 self)
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Partial answers are given to two questions. When does a lattice Λ contain a sublattice Λ ′ of index N that is geometrically similar to Λ? When is the sublattice “clean”, in the sense A similarity σ of norm c is a linear map from Rn to Rn such that σu · σv = c u · v for u,v ∈ Rn. Let Λ be an ndimensional rational lattice, i.e. u · v ∈ Q for u,v ∈ Λ. A sublattice Λ ′ ⊆ Λ is similar to Λ if σ(Λ) = Λ ′ for some similarity σ of norm c. We also call σ a multiplier of norm c for Λ. The index N = [Λ: Λ ′ ] is c n/2, so if n is odd c must be a square, say c = a 2,