Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids.

In this paper, we present several probabilistic transforms related to classical urn models. These transforms render the dependent random variables describing the urn occupancies into independent random variables with appropriate distributions. This simplifies the analysis of a large number of problems for which a function under investigation depends on the urn occupancies. The approach used for constructing the transforms involves generating functions of combinatorial numbers characterizing the urn distributions. We also show, by using Tauberian theorems derived in this paper, that under certain simple conditions the asymptotic expressions of target functions in the transform domain and in the inverse–transform domain are identical. Therefore, asymptotic information about certain statistics can be obtained without evaluating the inverse transform.

In this paper we prove several point selection theorems concerning objects ‘spanned’ by a finite set of points. For example, we show that for any set $P$ of $n$ points in $\R^2$ and any set $C$ of $m \,{\geq}\, 4n$ distinct pseudo-circles, each passing through a distinct pair of points of $P$, there is a point in $P$ that is covered by (i.e., lies in the interior of) $\Omega(m^2/n^2)$ pseudo-circles of $C$. Similar problems involving point sets in higher dimensions are also studied.

Most of our bounds are asymptotically tight, and they improve and generalize results of Chazelle, Edelsbrunner, Guibas, Hershberger, Seidel and Sharir [8], where weaker bounds for some of these cases were obtained.

What simple condition on a graph $G$ will ensure that $G\,{\succ}\,K_t$? As usual, $G\,{\succ}\,K_t$ means that $K_t$ is a minor of the graph $G$ (in other words, $G$ has vertex disjoint connected subgraphs $W_1,\ldots,W_t$ and at least one edge between $W_i$ and $W_j$, $1\,{\le}\, i<j\,{\le}\, t$).

How ‘tightly’ can we pack a given number of $r$-sets of an $n$-set? To be a little more precise, let $X=[n]=\{ 1,\ldots,n \}$, and let $X^r=\{ A\subset X : |A|=r \}$. For a set system $\mathcal{A}\subset X^r $, the neighbourhood of $\mathcal{A}$ is $N(\mathcal{A})=\{ B \in X^r: |B \bigtriangleup A|\le 2 \hbox{ for some }A \in \mathcal{A} \}$. In other words, $N(\mathcal{A})$ consists of those $r$-sets that are either in $\mathcal{A}$ or are ‘adjacent’ to it, in the sense that they are at minimal Hamming distance (i.e., distance 2) from some point of it. Given $|\mathcal{A}|$, how small can $|N(\mathcal{A})|$ be?

Heap ordered trees are planted plane trees, labelled in such a way that the labels always increase from the root to a leaf. We study two parameters, assuming that $p$ of the $n$ nodes are selected at random: the size of the ancestor tree of these nodes and the smallest subtree generated by these nodes. We compute expectation, variance, and also the Gaussian limit distribution, the latter as an application of Hwang's quasi-power theorem.

In this paper, we investigate the limit law of the inertial moment of Dyck paths with respect to the $x$-axis, that is, the sum of the squares of the altitudes. We find its Laplace transform using Louchard's methodology, rediscovering a result which was in fact well known by probabilists. We give recurrence relations which enable us to compute the moments of the joint limit law of the area and the inertial moments of both Dyck paths and Grand Dyck paths (bilateral Dyck paths).

In this paper, we show for generalized $M$-ary search trees that the Steiner distance of $p$ randomly chosen nodes in random search trees is asymptotically normally distributed. The special case $p=2$ shows, in particular, that the distribution of the distance between two randomly chosen nodes is asymptotically Gaussian. In the presented generating functions approach, we consider first the size of the ancestor-tree of $p$ randomly chosen nodes. From the obtained Gaussian limiting distribution for this parameter, we deduce the result for the Steiner distance. Since the size of the ancestor-tree is essentially the same as the number of passes in the (generalized) Multiple Quickselect algorithm, the limiting distribution result also holds for this parameter.

Let $F(\b{z})=\sum_\b{r} a_\b{r}\b{z^r}$ be a multivariate generating function that is meromorphic in some neighbourhood of the origin of $\mathbb{C}^d$, and let $\sing$ be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of $\sing$.

In the first article in this series, we treated the case of smooth points of $\sing$. In this article we deal with multiple points of $\sing$. Our results show that the central limit (Ornstein–Zernike) behaviour typical of the smooth case does not hold in the multiple point case. For example, when $\sing$ has a multiple point singularity at $(1, \ldots, 1)$, rather than $a_\b{r}$ decaying as $|\b{r}|^{-1/2}$ as $|\b{r}| \to \infty$, $a_\b{r}$ is very nearly polynomial in a cone of directions.

We show that, for a certain class of probabilistic models, the number of internal nodes $S_n$ of a trie built from $n$ independent and identically distributed keys is concentrated around its mean, in the sense that $\Var S_n=\Oh(\EE S_n)$. Keys are sequences of symbols which may be taken from varying alphabets, and the choice of the alphabet from which the $k$th symbol is taken, as well as its distribution, may depend on all the preceding symbols. In the construction of the trie we also allow for bucket sizes greater than 1. The property that characterizes our models is the following: there is a constant $C$ such that for any word $v$ that may occur as a prefix of a key the size $S^v_n$ of a trie built from the suffixes of $n$ independent keys conditioned to have the common prefix $v$ has the property $\EE S^v_n\leq Cn$. This class of models contains memoryless and Markovian source models as well as the probabilistic dynamical source models that were recently introduced and thoroughly developed by Vallée [Algorithmica29 (2001) 262–306], in particular the continued fraction source. Furthermore we study the external path length $L_n$, which obeys $\EE L_n=\Oh(n\ln n)$ and $\Var L_n=\Oh(n\ln^2 n)$.