We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science (automata and formal language theory), in particular, the rationality of generating functions of regular languages; (ii) analytic combinatorics that is used for deriving asymptotic properties from generating functions; (iii) computer algebra for determining generating functions explicitly, analysing generating functions and extracting coefficients efficiently. We provide constructions for overlapping or non-overlapping matches of a regular expression. A companion implementation produces multivariate generating functions for the statistics under study. A fast computation of Taylor coefficients of the generating functions then yields exact values of the moments with typical application to random t...

We show how asymptotic estimates of powers of polynomials with non-negative coefficients can be used in the analysis of low-density parity-check (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ensembles. We then consider the binary erasure channel (BEC). Using these estimates we derive lower bounds on the error exponent, under iterative decoding, of LDPC codes used over the BEC. Both regular and irregular code structures are considered. These bounds are compared to the corresponding bounds when optimal (maximum likelihood) decoding is applied.

The secondary structure of a RNA molecule is of great importance and possesses inuence, e.g. on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classication of secondary structures by means of their order proved useful with respect to numerous applications. In 1978 Waterman, who gave the rst precise formal framework for the topic, suggested to determine the number a n;p of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al. we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with Horton-Strahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a n;p . Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.

Given a multivariate generating function F (z1 ; : : : ; zd ) = P ar 1 ;:::;r d z r 1 1 z r d d , we determine asymptotics for the coecients. Our approach is to use Cauchy's integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers G treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.

We study the random variable Yn representing the number of occurrences of a symbol a in a word of length n chosen at random in a regular language L fa; bg where the random choice is de ned via a nonnegative rational formal series r of support L. Assuming that the transition matrix associated with r is primitive we obtain asymptotic estimates for the mean value and the variance of Yn and present a central limit theorem for its distribution. Under a further condition on such a matrix, we also derive an asymptotic approximation of the discrete Fourier transform of Yn that allows to prove a local limit theorem for Yn . Further consequences of our analysis concern the growth of the coecients in rational formal series; in particular, it turns out that, for a wide class of regular languages L, the maximum number of words of length n in L having the same number of occurrences of a given symbol is of the order of growth n , for some constant > 1.

We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful search is proportional to the number of ascendants of the sought element. On the other hand, the probabilistic behavior of the number of descendants is relevant for the analysis of paged data structures and for the analysis of the performance of quicksort, when recursive calls are not made on small subfiles. We also consider the number of ascendants and descendants of a random node in a random search tree, i.e., the grand averages of the quantities mentioned above. We address these questions for standard binary search trees and for locally balanced search trees. These search trees were introduced by Poblete and Munro and are binary search trees such that each subtree of size 3 is balanced; in oth...

Abstract Motivated by problems of pattern statistics, we study the limit distribution of the random variable counting the number of occurrences of the symbol ¥ in a word of length ¦ chosen at random in § ¥©¨����� � , according to a probability distribution defined via a finite automaton equipped with positive real weights. We determine the local limit distribution of such a quantity under the hypothesis that the transition matrix naturally associated with the finite automaton is primitive. Our probabilistic model extends the Markovian models traditionally used in the literature on pattern statistics. This result is obtained by introducing a notion of symbol-periodicity for irreducible matrices whose entries are polynomials in one variable over an arbitrary positive semiring. This notion and the related results we prove are of interest in their own right, since they extend classical properties of the Perron–Frobenius Theory for non-negative real matrices.

The Horton-Strahler number naturally arose from problems in various fields, e.g. geology, molecular biology and computer science. Consequently, detailed investigations of related parameters for different classes of binary tree structures are of interest. This paper shows one possibility of how to perform a mathematical analysis for parameters related to the Horton-Strahler number in a unified way such that only a single analysis is needed to obtain results for many different classes of trees. The method is explained by the examples of the expected Horton-Strahler number and the related r-th moments, the average number of critical nodes and the expected distance between critical nodes.

We survey recent work on the enumeration of non-crossing configurations on the set of vertices of a convex polygon, such as triangulations, trees, and forests. Exact formulae and limit laws are determined for several parameters of interest. In the second part of the talk we present results on the enumeration of chord diagrams (pairings of 2n vertices of a convex polygon by means of n disjoint pairs). We present limit laws for the number of components, the size of the largest component and the number of crossings. The use of generating functions and of a variation of Levy's continuity theorem for characteristic functions enable us to establish that most of the limit laws presented here are Gaussian. (Joint work by Marc Noy with Philippe Flajolet and others.) 1. Analytic Combinatorics of Non-crossing Configurations [3] 1.1. Connected Graphs and General Graphs. Let \Pi n = fv 1 ; : : : ; v n g be a fixed set of points in the plane, conventionally ordered counter-clockwise, that are verti...

Let L be an algebraic language on an alphabet X = fx 1 , x 2 , ..., x k g, and n a positive integer. We consider the problem of generating at random words of L with respect to a given distribution of the number of occurrences of the letters. We consider two alternatives of the problem. In the first one, a vector of natural numbers (n 1 , n 2 , ..., n k ) such that n 1 +n 2 + +n k = n is given, and the words must be generated uniformly among the set of words of L which contain exactly n i letters x i (1 i k). The second alternative consists, given v = (v 1 , ..., v k ) a vector of positive real numbers such that v 1 + + v k = 1, to generate at random words among the whole set of words of L of length n, in such a way that the expected number of occurrences of any letter x i equals nv i (1 i k), and two words having the same distribution of letters have the same probability to be generated. For this purpose, we design and study two alternatives of the recursive method which is classically employed for the uniform generation of combinatorial structures. This type of "controlled" non-uniform generation is of great interest in the statistical study of genomic sequences.