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Singular Combinatorics
 ICM 2002 VOL. III 13
, 2002
"... Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Sing ..."
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Cited by 387 (11 self)
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Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Singularity analysis" reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complexanalytic Tauberian procedure by which combinatorial constructions and asymptoticprobabilistic laws can be systematically related.
Monodromy of hypergeometric functions and nonlattice integral monodromy.Inst.HautesÉtudes Sci
 Publ. Math
"... The hypergeometric series (x) F(a,b;c;x) = Y, (a,n)(b,n) x".~0 (c,n) n! ' Ixl
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Cited by 110 (0 self)
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The hypergeometric series (x) F(a,b;c;x) = Y, (a,n)(b,n) x".~0 (c,n) n! ' Ixl<x nI
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 82 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Evaluations of kfold Euler/Zagier sums: a compendium of results for arbitrary k
 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 (NO.2) (1997), #R5
, 1997
"... Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble res ..."
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Cited by 69 (28 self)
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Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.
Crossings and nestings of matchings and partitions
 Trans. Amer. Math. Soc
"... Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number ..."
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Cited by 61 (15 self)
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Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of knonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no kcrossing (or with no knesting). 1.
Lectures on non perturbative field theory and quantum impurity problems, in Topological aspects of low dimensional systems (Les Houches
, 1998
"... They are a sequel to the notes I wrote two years ago for the Summer School, “Topological Aspects ..."
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Cited by 47 (1 self)
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They are a sequel to the notes I wrote two years ago for the Summer School, “Topological Aspects
Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
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Cited by 46 (9 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".
Ramanujan’s theories of elliptic functions to alternative bases, Trans.Amer.Math.Soc.,347
, 1995
"... Abstract. In his famous paper on modular equations and approximations to n, Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories " in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recent ..."
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Cited by 44 (15 self)
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Abstract. In his famous paper on modular equations and approximations to n, Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories " in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.