An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

This paper examines the extremal problem of how many 1-entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of n-permutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut [1]. 1

. Consider, for a permutation oe 2 S k , the number F (n; oe) of permutations in Sn which avoid oe as a subpattern. The conjecture of Stanley and Wilf is that for every oe there is a constant c(oe) ! 1 such that for all n, F (n; oe) c(oe) n . All the recent work on this problem also mentions the "stronger conjecture" that for every oe, the limit of F (n; oe) 1=n exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity. We also discuss n-permutations, containing all oe 2 S k as subpatterns. We prove that this can be achieved with n = k 2 , we conjecture that asymptotically n (k=e) 2 is the best achievable, and we present Noga Alon's conjecture that n (k=2) 2 is the threshold for random permutations. Mathematics Subject Classification: 05A05,05A16. 1. Introduction Consider, for a permutation oe 2 S k , the set A(n; oe) of permutations 2 S n which avoid oe as a subpattern, and it...

Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 <... < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = |{τ ∈ Sn | τ avoids σ}|. Stanley and Wilf conjectured that for any σ ∈ Sk there exists a constant c = c(σ) such that F (n, σ) ≤ c n for all n. Here we prove the following weaker statement: For every fixed σ ∈ Sk, F (n, σ) ≤ c nγ ∗ (n) , where c = c(σ) and γ ∗ (n) is an extremely slow growing function, related to the Ackermann hierarchy. 1

Abstract. We show that the Stanley–Wilf enumerative conjecture on permutations follows easily from the Füredi–Hajnal extremal conjecture on 0-1 matrices. We apply the method, discovered by Alon and Friedgut, that derives an (almost) exponential bound on the number of some objects from a (almost) linear bound on their sizes. They proved by it a weaker form of the Stanley–Wilf conjecture. Using bipartite graphs, we give a simpler proof of their result. Pokaжem, qto gipoteza Stзnli i Vilfa o qisle perestanovok vytekaet prostym obrazom iz зkstremalьno � gipotezy Firedy i Ha�nala o 0-1 matricah. Primen�em metod vyvoda (poqti) зksponencialьno� ocenki qisla obъektov iz (poqti) line�no � ocenki ih veliqin otkryty � Alonom i Fridgutom. Зtim metodom oni dokazali gipotezu Stзnli i Vilfa v oslablenno � forme. S pomowь� dvudolьnyh grafov poluqim bolee prostoe dokazatelьstvo ih rezulьtata. The Stanley–Wilf conjecture asserts that the number of n-permutations not

A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and the number of such Hs is O(c n). The latter part generalizes the Stanley--Wilf conjecture on permutations. Using generalized Davenport--Schinzel sequences, we prove the conjectures with weaker bounds O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p

The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nesetril) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u; n) and we present also two new theorems which give good upper bounds on Ex(u; n) for u consisting of (two) smaller subsequences u i provided we have good upper bounds on Ex(u i ; n). We use these theorems to describe a wide class of sequences u ("linear sequences") for which Ex(u; n) = O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u; n).

We survey in detail...

To Dan Kleitman, on his birthday, with all good wishes. Let n, k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. 1 We will call τ the pattern. We will look for the pattern τ in permutations σ of n letters. A pattern τ is said to occur in a permutation σ if there are integers 1 ≤ i1 <i2 <...<ik ≤ n such that for all 1 ≤ r<s ≤ k we have τ(r) <τ(s) if and only if σ(ir) <σ(is). Example: Suppose τ = (132). Then this pattern of k = 3 letters occurs several times in the following permutation σ, ofn = 14 letters (one such occurrence is underlined): σ =(529414 10 13615811 7 13 12) 1 Some areas of research and recent results Among the active areas of research are the following. 1. For a given pattern τ, let f(n, τ) be the number of τ-free permutations of n letters. Describe the equivalence classes of patterns that have the same f. 2. What can be said about the asymptotics of f(n, τ) for n →∞and fixed τ? 3. For fixed τ what is the maximum number of occurrences of τ in a permutation of n letters? Call this g(τ,n). Which permutation has the maximum? 1 We will often refer to {1, 2,...,k} as the letters on which the permutation acts, however their numerical sizes will be very relevant. 1 2 Packing density First, as regards the question of stuffing in as many τ’s as possible, Fred Galvin (unpublished) has shown the following. Theorem 1 (Galvin) For fixed τ ∈ Sk, is decreasing, and thus exists. � � ∞ g(τ,n) �n � k lim n→∞ n=k g(τ,n) Galvin’s proof is reproduced here with his permission: Let τ be a fixed pattern of length k. If x is any sequence of distinct numbers, of length ≥ k, let g(x) be the number of τ-subsequences of x, and let h(x) =g(x) / � |x | � k.Forn≥k, let f(n, τ) H(n) = max {h(x):|x | = n} = �. Suppose k ≤ m<n, and let x be a permutation of length n with h(x) =H(n). Note that h(x) is equal to the average of h(y) over all m-termed subsequences y of x, and therefore cannot exceed

We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a Davenport-Schinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverse-Ackermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.