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An instance of the square packing problem consists of n squares with independently, uniformly distributed side-lengths and independently, uniformly distributed locations on the unit d-dimensional torus. A packing is a maximum family of pairwise disjoint squares. The one-dimensional version of the problem is the classical random interval packing problem. This paper deals with the asymptotic behaviour of packings as n tends to infinity while d = 2. Coffman, Lueker, Spencer and Winkler recently proved that the average size of packing is Θ(nd/(d+1)). Using partitioning techniques, sub-additivity and concentration of measure arguments, we show first that, after normalization by n2/3, the size of two-dimensional square packings tends in probability toward a genuine limit γ. Straightforward concentration arguments show that large fluctuations of order n2/3 should have probability vanishing exponentially fast with n2/3. Even though γ remains unknown, using a change of measure argument we show that this upper bound on tail probabilities is qualitatively correct.
We define a space of random edge-coloured graphs [Gscr]n,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 [les ] 21 such that, provided m [ges ] K0n log n and κ [ges ] K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.
We show that, for every positive integer c*, there is an integer n such that, if M is a matroid whose largest cocircuit has size c*, then E(M) can be partitioned into two sets E1 and E2 such that every connected component of each of M[mid ]E1 and M[mid ]E2 has at most n elements.
Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then
This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).
Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show
Let δ be the minimal degree of G. We show
This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.
How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.
For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).
Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.
We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an ‘interaction graph’. During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.
We give a concentration inequality involving a family of independent random permutations, which is useful for analysing certain randomized methods for graph colouring.
Combinatorics on words is a field that has grown separately within several branches of mathematics, such as number theory, group theory or probability theory, and appears frequently in problems of theoretical computer science, as dealing with automata and formal languages.
A unified treatment of the theory appeared in Lothaire's Combinatorics on Words. Since then, the field has grown rapidly. This book presents new topics of combinatorics on words.
Several of them were not yet ripe for exposition, or even not yet explored, twenty years ago. The spirit of the book is the same, namely an introductory exposition of a field, with full proofs and numerous examples, and further developments deferred to problems, or mentioned in the Notes.
This book is independent of Lothaire's first book, in the sense that no knowledge of the first volume is assumed. In order to avoid repetitions, some results of the first book, when needed here, are explicitly quoted, and are only referred for the proof to the first volume.
This volume presents, compared with the previous one, two important new features. It is first of all a complement in the sense that it goes deeper in the same direction. For example, the theory of unavoidable patterns (Chapter 3) is a generalization of the theory of square-free words and morphisms. In the same way, the chapters on statistics on words and permutations (Chapters 10 and 11) are a continuation of the chapter on transformations on words of the previous volume.
The aim of this chapter is to provide an introduction to several concepts used elsewhere in the book. It fixes the general notation on words used elsewhere. It also introduces more specialized notions of general interest. For instance, the notion of a uniformly recurrent word used in several other chapters is introduced here.
We start with the notation concerning finite and infinite words. We also describe the Cantor space topology on the space of infinite words.
We provide a basic introduction to the theory of automata. It covers the determinization algorithm, part of Kleene's theorem, syntactic monoids and basic facts about transducers. These concepts are illustrated on the classical combinatorial examples of the de Bruijn graph, and the Morse-Hedlund theorem.
We also consider the relationship with generating series, as a useful tool for the enumeration of words.
We introduce some basic concepts of symbolic dynamical systems, in relation with automata. We prove the equivalence between the notions of minimality and uniform recurrence. Entropy is considered, and we show how to compute it for a sofic system.
We also present a more specialized subject, namely unavoidable sets. This notion is easy to define but leads to interesting and significant results. In this sense, the last section of this chapter is a foretaste of the rest of the book.
In this chapter we briefly summarize some of the important results in information theory which we have not been able to treat in detail. We shall give no proofs, but instead refer the interested reader elsewhere, usually to a textbook, sometimes to an original paper, for details.
We choose to restrict our attention solely to generalizations and extensions of the twin pearls of information theory, Shannon's channel coding theorem (Theorem 2.4 and its corollary) and his source coding theorem (Theorem 3.4). We treat each in a separate section.
The channel coding theorem
We restate the theorem for reference (see Corollary to Theorem 2.4).
Associated with each discrete memoryless channel, there is a nonnegative number C (called channel capacity) with the following property. For any ε > 0 and R < C, for large enough n, there exists a code of length n and rate ≧ R (i.e., with at least 2Rn distinct codewords), and an appropriate decoding algorithm, such that, when the code is used on the given channel, the probability of decoder error is < ε.
We shall now conduct a guided tour through the theorem, pointing out as we go places where the hypotheses can be weakened or the conclusions strengthened. The points of interest will be the phrases discrete memoryless channel, a nonnegative number C, for large enough n and there exists a code … and … decoding algorithm. We shall also briefly discuss various converses to the coding theorem.
This chapter is devoted, as was the previous one, to combinatorial properties of permutations, considered as words. The starting point in this subject is the bivalent status of permutations, which can be considered as products of cycles as well as a sequence of the first n integers written in disorder.
The fundamental results concerning this area are presented in Chapter 10 of Lothaire 1983. They consist essentially in two transformations on words. The first one (first fundamental transformation) is an encoding of the cycle decomposition of a permutation. The second one (second fundamental transformation) accounts for a statistical property of permutations, namely the equidistribution of the number of inversions and the inverse major index on permutations with a given shape.
In the previous chapter (Chapter 10) some other properties are presented, including basic facts on q-calculus and additional statistics on permutations.
In this chapter, we carry on with complements focused on two main aspects. The first one is a shortcut avoiding the second fundamental transformation by a simple evaluation of determinants. The second one is the analogy between the first fundamental transformation and the Lyndon factorization (the Gessel normalization).
The organization of the chapter is the following. After some preliminaries, Section 11.2 provides a determinantal expression for the commutative image of some sets of words. This expression is used in Sections 11.3 and 11.4, for evaluating respectively the inverse major index and the number of inversions of permutations with a given shape. In Section 11.5 the Gessel normalization is introduced. In Section 11.6, it is then applied to evaluating the major index of permutations with a given cycle structure.
Sturmian words are infinite words over a binary alphabet that have exactly n + 1 factors of length n for each n ≥ 0. It appears that these words admit several equivalent definitions, and can even be described explicitly in arithmetic form. This arithmetic description is a bridge between combinatorics and number theory. Moreover, the definition by factors makes Sturmian words define symbolic dynamical systems. The first detailed investigations of these words were done from this point of view. Their numerous properties and equivalent definitions, and also the fact that the Fibonacci word is Sturmian, have led to a great development, under various terminologies, of the research.
The aim of this chapter is to present basic properties of Sturmian words and of their transformation by morphisms. The style of exposition relies basically on combinatorial arguments.
The first section is devoted to the proof of the Morse–Hedlund theorem stating the equivalence of Sturmian words with the set of balanced aperiodic words and the set of mechanical words of irrational slope. We also mention several other formulations of mechanical words, such as rotations and cutting sequences. We next give properties of the set of factors of one Sturmian word, such as closure under reversal, the minimality of the associated dynamical system, the fact that the set depends only on the slope, and we give the description of special words.
At the beginning of Chapter 7, we said that by restricting our attention to linear codes (rather than arbitrary, unstructured codes), we could hope to find some good codes which are reasonably easy to implement. And it is true that (via syndrome decoding, for example) a “small” linear code, say with dimension or redundancy at most 20, can be implemented in hardware without much difficulty. However, in order to obtain the performance promised by Shannon's theorems, it is necessary to use larger codes, and in general, a large code, even if it is linear, will be difficult to implement. For this reason, almost all block codes used in practice are in fact cyclic codes; cyclic codes form a very small and highly structured subset of the set of linear codes. In this chapter, we will give a general introduction to cyclic codes, discussing both the underlying mathematical theory (Section 8.1) and the basic hardware circuits used to implement cyclic codes (Section 8.2). In Section 8.3 we will show that Hamming codes can be implemented as cyclic codes, and in Sections 8.4 and 8.5 we will see how cyclic codes are used to combat burst errors. Our story will be continued in Chapter 9, where we will study the most important family of cyclic codes yet discovered: the BCH/ Reed–Solomon family.
We begin our studies with the innocuous-appearing definition of the class of cyclic codes.
Introduction: The generator and parity-check matrices
We have already noted that the channel coding Theorem 2.4 is unsatisfactory from a practical standpoint. This is because the codes whose existence is proved there suffer from at least three distinct defects:
(a) They are hard to find (although the proof of Theorem 2.4 suggests that a code chosen “at random” is likely to be pretty good, provided its length is large enough).
(b) They are hard to analyze. (Given a code, how are we to know how good it is? The impossibility of computing the error probability for a fixed code is what led us to the random coding artifice in the first place!)
(c) They are hard to implement. (In particular, they are hard to decode: the decoding algorithm sugggested in the proof of Theorem 2.4—search the region S(y) for codewords, and so on—is hopelessly complex unless the code is trivially small.)
In fact, virtually the only coding scheme we have encountered so far which suffers from none of these defects is the (7, 4) Hamming code of the Introduction. In this chapter we show that the Hamming code is a member of a very large class of codes, the linear codes, and in Chapters 7–9 we show that there are some very good linear codes which are free from the three defects cited above.