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 complex-analytic Tauberian procedure by which combinatorial constructions and asymptotic-probabilistic laws can be systematically related.

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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.

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.

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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 k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting). 1.

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this paper is to establish all of these claims. In proving these results, it is very clear to us that Ramanujan had established further results that he unfortunately did not record either in his notebooks and other unpublished papers or in his published papers. Moreover, Ramanujan's work points the way to many additional theorems in these theories, and we hope that others will continue to develop Ramanujan's beautiful ideas. The most important of the three alternative theories is the one arising from the hypergeometric function 2 F 1

Approximate counting is an algorithm proposed by R. Morris which makes it possible to keep approximate counts of large numbers in small counters. The algorithm is useful for gathering statistics of a large number of events as well as for applications related to data compression (Todd et al.). We provide here a complete analysis of approximate counting which establishes good convergence properties of the algorithm and allows to quantify precisely complexity-accuracy tradeoffs. Introduction. As shown by an easy information-theoretic argument, maintaining a counter whose values may range in the interval 1 to M essentially necessitates log,M bits. This lower bound is of course achieved by a 1 standard binary counter. R. Morris [8] has proposed a probabilistic algorithm that maintains an

Let S be a smooth projective surface and S [n] the Hilbert scheme of zerodimensional subschemes of S of length n. We proof that the class of S [n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characteristics of certain tautological sheaves on S [n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves. Let S be a smooth projective surface over the field of complex numbers. For a nonnegative integer n let S [n] denote the Hilbert scheme parameterizing zerodimensional subschemes of length n. By a well-known result of Fogarty [10] the scheme S [n] is smooth and projective of dimension 2n, and is irreducible if S is irreducible. Let Ω = Ω U ⊗ Q be the complex cobordism ring with rational coefficients. Milnor [20] showed that Ω is a polynomial ring freely generated by the cobordism classes [CPi] for i ∈ N. For a smooth and projective complex surface we define