Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process d j ≤ B balls are placed in distinct urns with uniform probability. Let M i (j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M 0(j) and M 1(j) is obtained. A multivariate generating function for the joint factorial moments of M i (j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the d j , j ≥ 1, are independent, identically distributed random variables is investigated.

Consider the random variable L n defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=limn→∞E[L n ]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (q m )m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂m , of the q m such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

We propose a simple and efficient scheme for ranking all teams in a tournament where matches can be played simultaneously. We show that the distribution of the number of rounds of the proposed scheme can be derived using lattice path counting techniques used in ballot problems. We also discuss our method from the viewpoint of parallel sorting algorithms.

In DNA sequences, specific words may take on biological functions as marker or signalling sequences. These may often be identified by frequent-word analyses as being particularly abundant. Accurate statistics is needed to assess the statistical significance of these word frequencies. The set of shuffled sequences - letter sequences having the same k-word composition, for some choice of k, as the sequence being analysed - is considered the most appropriate sample space for analysing word counts. However, little is known about these word counts. Here we present exact formulae for word counts in shuffled sequences.

Let Y k (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0 Y k (ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Y k . We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Y k (ω) := ∑j=0 k Y j (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Y k k , and show that, as n → ∞, Y k /n k is asymptotically Gaussian under the conditional distribution P(· | Z = n).

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.

We investigate the limit distributions associated with cost measures in Sattolo's algorithm for generating random cyclic permutations. The number of moves made by an element turns out to be a mixture of 1 and 1 plus a geometric distribution with parameter ½, where the mixing probability is the limiting ratio of the rank of the element being moved to the size of the permutation. On the other hand, the raw distance traveled by an element to its final destination does not converge in distribution without norming. Linearly scaled, the distance converges to a mixture of a uniform and a shifted product of a pair of independent uniforms. The results are obtained via randomization as a transform, followed by derandomization as an inverse transform. The work extends analysis by Prodinger.

We derive the asymptotic joint normality, by a martingale approach, for the numbers of upper records, lower records and inversions in a random sequence.

We give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral Partition Functions.

The application of the generalised ballot theorem to queueing theory leads to elegant results for the simple M/G/1 queue. It is thought that such results are not possible for more general M/G/1-type queues. We, however, derive a batch ballot theorem which can be applied to derive the first passage distribution matrix, G , for the general M/G/1-type queue.

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

We analyse the expected performance of various group testing, or pooling, designs. The context is that of identifying characterized clones in a large collection of clones. Here we choose as performance criterion the expected number of unresolved ‘negative’ clones, and we aim to minimize this quantity. Technically, long inclusion–exclusion summations are encountered which, aside from being computationally demanding, give little inkling of the qualitative effect of parametric control on the pooling strategy. We show that readily-interpreted re-summation can be performed, leading to asymptotic forms and systematic corrections. We apply our results to randomized designs, illustrating how they might be implemented for approximating combinatorial formulae.

A random mapping (T n ;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G T representing (T n ;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

A general analytic scheme for Poisson approximation to discrete distributions is studied in which the asymptotic behaviours of the generalized total variation, Fortet-Mourier (or Wasserstein), Kolmogorov and Matusita (or Hellinger) distances are explicitly characterized. Applications of this result include many number-theoretic functions and combinatorial structures. Our approach differs from most of the existing ones in the literature and is easily amended for other discrete approximations; arithmetic and combinatorial examples for Bessel approximation are also presented. A unified approach is developed for deriving uniform estimates for probability generating functions of the number of components in general decomposable combinatorial structures, with or without analytic continuation outside their circles of convergence.

Given a group G and a finite generating set G, we take pG: G → Z to be the function which counts the number of geodesics for each group element g. This generalizes Pascal's triangle. We compute pG for word hyperbolic and describe generic behaviour in abelian groups.

A new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.