The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.

There are at least four U.S. patents on software watermarking, and an idea for further advancing the state of the art was presented in 1999 by Collberg and Thomborsen. The new idea is to embed a watermark in dynamic data structures, thereby protecting against many programtransformation attacks. Until now there have been no reports on practical experience with this technique. We have implemented and experimented with a watermarking system for Java based on the ideas of Collberg and Thomborsen. Our experiments show that watermarking can be done efficiently with moderate increases in code size, execution times, and heap-space usage, while making the watermarked code resilient to a variety of programtransformation attacks. For a particular representation of watermarks, the time to retrieve a watermark is on the order of one minute per megabyte of heap space. Our implementation is not designed to resists all possible attacks; to do that it should be combined with other protection techniques such as obfuscation and tamperproofing.

Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n ? 1, m ? 2, and let oe 0 be a permutation of Sn having d i cycles of length i, for i ? 1. We prove that the number of m-tuples (oe 1 ; : : : ; oe m ) of permutations of Sn such that: ffl oe 1 oe 2 \Delta \Delta \Delta oe m = oe 0 , ffl the group generated by oe 1 ; : : : ; oe m acts transitively on f1; 2; : : : ; ng, ffl P m i=0 c(oe i ) = n(m \Gamma 1) + 2, where c(oe i ) denotes the number of cycles of oe i , is m [(m \Gamma 1)n \Gamma 1]! [(m \Gamma 1)n \Gamma c(oe0 ) + 2]! Y i?1 " i / mi \Gamma 1 i !# d i : A one-to-one correspondence relates these m-tuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m = 2, we recover a formula of Tutte for the number of Eulerian maps. The proof extends the method applied in [21] to the latter case, and relies on the idea...

We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τ-function for the Toda lattice hierarchy of Ueno and Takasaki.

Abstract. Let {ar: r ∈ Nd} be a d-dimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have

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1.1. Recursions and Gromov-Witten theory 2