We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

In 2011, shortly after his untimely passing, several colleagues and I wrote an obituary for Philippe Flajolet [5] that includes the following:

We consider the languages of finite trees called tree-shift languages which are factorial extensible tree languages. These languages are sets of factors of subshifts of infinite trees. We give effective syntactic characterizations of two classes of regular tree-shift languages: the finite type tree languages and the tree languages which are almost of finite type. Each class corresponds to a class of subshifts of trees which is invariant by conjugacy. For this goal, we define a tree algebra which is finer than the classical syntactic tree algebra based on contexts. This allows us to capture the notion of constant tree which is essential in the framework of tree-shift languages.

We investigate the finite repetition threshold for k-letter alphabets, k ≥ 4, that is the smallest number r for which there exists an infinite r+-free word containing a finite number of r-powers. We show that there exists an infinite Dejean word on a 4-letter alphabet (i.e. a word without factors of exponent more than 7/5) containing only two 7/5-powers. For a 5-letter alphabet, we show that there exists an infinite Dejean word containing only 60 5/4-powers, and we conjecture that this number can be lowered to 45. Finally we show that the finite repetition threshold for k letters is equal to the repetition threshold for k letters, for every k ≥ 6.

We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be Θ(n4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov [Problemy Kibernet. 39 (1982) 67–84], then independently by Mignosi [Theoret. Comput. Sci. 82 (1991) 71–84], and others.

A k-abeliancube is a word uvw, where the factors u, v, and w are either pairwiseequal, or have the same multiplicities for every one of their factors of length at mostk.Previously it has been shown that k-abelian cubes are avoidable over a binaryalphabet for k ≥8. Here it is proved that this holds for k ≥ 5.

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.

In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableaux may have arbitrary positive weights. (We consider only the case with the parameters u = q = 1.) Under this general model we derive a number of results. Some of our results concern the limiting laws for the number of appearances of symbols in a random staircase tableaux. They generalize and subsume earlier results that were obtained for specific values of the weights.

One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature. Furthermore, our generality allows us to analyse the structure of random staircase tableaux, and we obtain several results in this direction.

One of the tools we use is the generating functions of the parameters of interest. This leads us to a two-parameter family of polynomials, generalizing the classical Eulerian polynomials.

We also briefly discuss the relation of staircase tableaux to the asymmetric exclusion process, to other recently introduced types of tableaux, and to an urn model studied by a number of researchers, including Philippe Flajolet.

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.

If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less than N. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Laminations are classic sets of disjoint and non-self-crossing curves on surfaces.Lamination languages are languages of two-way infinite words which code laminations byusing associated labeled embedded graphs, and which are subshifts. Here, we characterizethe possible exact affine factor complexities of these languages through bouquets ofcircles, i.e. graphs made of one vertex, as representative coding graphs.We also show how to build families of laminations together with corresponding laminationlanguages covering all the possible exact affine complexities.

For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1+\sqrt 5)/2$ ([18]).

We resolve a conjecture proposed by D.E. Knuth concerning a recurrence arising in the satisfiability problem. Knuth's recurrence resembles recurrences arising in the analysis of tries, in particular PATRICIA tries, and asymmetric leader election. We solve Knuth's recurrence exactly and asymptotically, using analytic techniques such as the Mellin transform and analytic depoissonization.

We define the notion of smooth supercritical compositional structures. Two well-known examples are compositions and graphs of given genus. The ‘parts’ of a graph are the subgraphs that are maximal trees. We show that large part sizes have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. In many cases this leads to asymptotic formulas for the probability of being gap-free and for the expected values of the largest part sizes, number of distinct parts and number of parts of multiplicity k.

Let ${\cal A}$ be a minor-closed class of labelled graphs, and let ${\cal G}_{n}$ be a random graph sampled uniformly from the set of n-vertex graphs of ${\cal A}$. When n is large, what is the probability that ${\cal G}_{n}$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.

Using exact enumeration, we study a collection of classes ${\cal A}$ excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function C(z) that counts connected graphs of ${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

We define a sequence of tree-indexed processes closely related to the operation of the QuickSelect search algorithm (also known as Find) for all the various values of n (the number of input keys) and m (the rank of the desired order statistic among the keys). As a ‘master theorem’ we establish convergence of these processes in a certain Banach space, from which known distributional convergence results as n → ∞ about

1. (1) the number of key comparisons required

   are easily recovered

   1. (a) when m/n → α ∈ [0, 1], and

   2. (b) in the worst case over the choice of m.

   From the master theorem it is also easy, for distributional convergence of

2. (2) the number of symbol comparisons required,

Our techniques allow us to unify the treatment of cases (1) and (2) and indeed to consider many other cost functions as well. Further, all our results provide a stronger mode of convergence (namely, convergence in Lp or almost surely) than convergence in distribution. Extensions to MultipleQuickSelect are discussed briefly.

We consider the problem of enumerating d-irreducible maps, i.e., planar maps all of whose cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approach via substitution, where these maps are obtained from general maps by a replacement of all d-cycles by elementary faces, and a bijective approach via slice decomposition, which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions of d-irreducible maps with controlled face degrees, summarized in some elegant ‘pointing formula’. We provide an equivalent description of d-irreducible slices in terms of so-called d-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.

Formal grammars such as L-systems have long been used to describe plant growth dynamics. In this article, they are used for a new purpose. The aim is to build a symbolic method that enables the computation of the stochastic distribution associated with the number of complex structures in plants whose organogenesis is driven by a multitype branching process. For that purpose, a new combinatorial framework is set in which plant structure is coded by a Dyck word. Moreover, organogenesis is represented by stochastic F0L-systems. In doing so, the problem is equivalent to determining the distribution of patterns in random words generated by a stochastic F0L-system. This method finds interesting applications in the parameter identification of stochastic models of plant development.