Abstract: In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

Abstract: The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose–Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.

Abstract: We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Abstract: Let H d (n,p) signify a random d -uniform hypergraph with n vertices in which each of the $\binom{n}{d}$ possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d -uniform hypergraph with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of H d (n,p) and H d (n,m) in the regime $(d-1)\binom{n-1}{d-1}p>1+\varepsilon$ , resp. d(d−1)m/n >1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach.

Abstract: olya 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 P ´ olya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

Abstract: In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011).We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length O(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank.At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

Abstract: We prove that the number of 1324-avoiding permutations of length n is less than $(7+4\sqrt{3})^n$ . The novelty of our method is that we injectively encode such permutations by a pair of words of length n over a finite alphabet that avoid a given factor.

Abstract: In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P (x)F (x), where P (x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F (x) around x = 1. Numerous examples from the literature, as well as some new statistics are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics. In the analysis of partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) = ∞ Y j=1 (1 − x j ) −1 is the generating function for the number of partitions. In this paper, we want to develop a general asymptotic scheme that allows one to derive an asymptotic formula for the n-th coefficient of P(x)F(x) from the behaviour of F(x) as x → 1. It is well known that p(n) = (x n )P(x) essentially behaves like 1 4 √ 3n exp � π p 2n/3 �

Abstract: We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all 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.

Abstract: We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,\ldots,k$ are in distinct cycles of the random permutation of $\{1,2,\ldots,n\}$ obtained as product of two uniformly random $n$-cycles.

Abstract: We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n tends to infinity, for the proportion of visited vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton-Watson tree with respect to the success epochs of the coupon collector problem.

Abstract: A rational polyhedron$P\\subseteq {\\mathbb{R^n}}$ is a finite union of simplexes in ${\\mathbb{R^n}}$ with rational vertices. P is said to be $\\mathbb Z$-homeomorphic to the rational polyhedron $Q\\subseteq {\\mathbb{R^{\\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\\mathbb Z$-homeomorphism amounts to continuous $\\mathcal{G}_n$-equidissectability, where $\\mathcal{G}_n=GL(n,\\mathbb Z) \\ltimes \\mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\\mathbb{R^{n}}$ that leave the lattice $\\mathbb Z^{n}$ invariant. $\\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\\mathbb Z$-homeomorphic rational polyhedra $$P\\subseteq {\\mathbb{R^n}}$$ and $Q\\subseteq {\\mathbb{R^{\\it m}}}$ satisfy $\\lambda_d(P)=\\lambda_d(Q)$. $\\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\\mathbb{R^{\\it n}}, $\\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

Abstract: 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.

Abstract: We study a discrete time self interacting random process on graphs, which we call Greedy Random Walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not been crossed yet by the walker. At each step, the walker being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walk jumps along it to the neighboring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\Z^d$ for all $d \geq 3$.

Abstract: In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable Xk. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n.A few years ago, Bassetti and Diaconis showed that, for a sampler a la Knuth that generates walks crossing a k × k square and consisting of North and East steps, the relative variance is only . This is exponential in the average length of the walks, which is of order k2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k2 (which is again the average length of these walks).Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following the paper by Bassetti and Diaconis, is one of very few examples where the variance of the estimator can be found.

Abstract: We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-e)Δt for graphs of maximum degree at most Δ, where e is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

Abstract: Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [26] proved that for such rules, there exists a (rule-dependent) critical time tc such that when t tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n2γ log4n, on the size of the largest component, at time instants tn = tc−n−γ, where γ ∈ (0,1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behaviour of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space , equipped with the usual Skorokhod topology. The coupling construction also gives an alternative characterization (from the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces.

Abstract: We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g(y−x), where g is a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

Abstract: For d ≥ 2, let Hd(n,p) denote a random d-uniform hypergraph with n vertices in which each of the $\\binom{n}{d}$ possible edges is present with probability p=p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. Let either H=Hd(n,m) or H=Hd(n,p), where m/n and $\\binom{n-1}{d-1}p$ need to be bounded away from (d−1)−1 and 0 respectively. We determine the asymptotic probability that H is connected. This yields the asymptotic number of connected d-uniform hypergraphs with given numbers of vertices and edges. We also derive a local limit theorem for the number of edges in Hd(n,p), conditioned on Hd(n,p) being connected.

Abstract: For any c ≥ 2, a c-strong colouring of the hypergraph G is an assignment of colours to the vertices of G such that, for every edge e of G, the vertices of e are coloured by at least min{c,|e|} distinct colours. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common.A natural variant of a question of Erdős and Lovasz is: For fixed c ≥ 2 and t ≥ 1, what is the minimum number of colours that is sufficient to c-strong colour any t-intersecting hypergraphs? The purpose of this note is to describe some open problems related to this question.

Abstract: Given an edge colouring of a graph with a set of m colours, we say that the graph is (exactly) m-coloured if each of the colours is used. We consider edge colourings of the complete graph on N with innitely many colours and show that either one can nd an m-coloured complete subgraph for every natural number m or there exists an innite subset X N coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that Xnfvg is 1-coloured and each edge between v and Xnfvg has a distinct colour (all dierent to the colour used on Xnfvg). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about nding m-coloured complete subgraphs in colourings with nitely many colours.

Abstract: We propose a counting dimension for subsets of $\mathbb{Z}$ and prove that, under certain conditions on E,F ⊂ $\mathbb{Z}$ , for Lebesgue almost every λ ∈ $\mathbb{R}$ the counting dimension of E + ⌊λ F ⌋ is at least the minimum between 1 and the sum of the counting dimensions of E and F . Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E + ⌊λ F ⌋ has positive upper Banach density for Lebesgue almost every λ ∈ $\mathbb{R}$ . The result has direct consequences when E,F are arithmetic sets, e.g ., the integer values of a polynomial with integer coefficients.

Abstract: We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Haggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size. © Cambridge University Press 2013.

Abstract: Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that 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 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 dominant singularity of the generating function C(z) that counts connected graphs of 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 deserves a systematic investigation.

Abstract: An intracellular hallmark of Alzheimer Disease (AD) is accumulation of hyperphosphorylated tau as tangles of paired helical filaments (PHF). A significant advance in understanding tau's behaviour isolated came when it was recognized that the protein contains isolated short peptide motifs, embedded in an otherwise hydrophilic environment, which have a high tendency for beta-structure and aggregation, forming the core of the PHF. In a recent work, we used the smallest fragment responsible for aggregation, the hexapeptide 306 VQIVYK 311 , in order to investigate with molecular dynamics simulations possible binding modes of the tau protein fragment with respect to an active flavonoid, which would be responsible for the inhibitory process of aggregation of tau. Considering such results, we have used in this work a selected pharmacophoric model and carried out a pharmacophore-based virtual screening with the purpose of designing novel potential Tau aggregation inhibitors. An initial set of 96 compounds was selected, of which 86 are unpublished regarding Tau anti-aggregation activity and the other 10 compounds are reported as Tau ligands. Prediction of biological activity and pharmaceutical properties indicated four tiophene derivatives as promising Tau aggregation inhibitors for Alzheimer's disease treatment.

Abstract: We resolve a conjecture proposed by DE 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