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45
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel 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 noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 471 (115 self)
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An (n; s) DavenportSchinzel 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 noncontiguous) 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 DavenportSchinzel 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 nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Excluded permutation matrices and the StanleyWilf conjecture
 J. Combin. Theory Ser. A
, 2004
"... This paper examines the extremal problem of how many 1entries 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 set ..."
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Cited by 119 (4 self)
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This paper examines the extremal problem of how many 1entries 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 npermutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut [1]. 1
On the StanleyWilf conjecture for the number of permutations avoiding a given pattern
, 1999
"... . 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 ..."
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Cited by 61 (0 self)
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. 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 npermutations, 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...
The FürediHajnal conjecture implies the StanleyWilf conjecture
 Formal Power Series and Algebraic Combinatorics
, 2000
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On the number of permutations avoiding a given pattern
 J. Comb. Theory, Ser. A
, 1999
"... 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 ..."
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Cited by 42 (0 self)
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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
Improved bounds and new techniques for DavenportSchinzel sequences and their generalizations
 In Proceedings 20th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2009
"... We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes th ..."
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Cited by 29 (1 self)
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We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (1989), had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lowerorder terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λs(n). This new technique is based on some recurrences very similar to those used by the author, together with Alon, Kaplan, Sharir, and Smorodinsky (SODA 2008), for the problem of stabbing interval chains with jtuples. With this new technique we: (1) rederive the upper bound of λ3(n) ≤ 2nα(n)+O ( n √ α(n) ) (first shown by Klazar, 1999); (2) rederive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport–Schinzel sequences considered by Adamec, Klazar, and Valtr (1992). Regarding lower bounds, we show that λ3(n) ≥ 2nα(n) − O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1 2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds of λs(n) ≥ n·2 (1/t!)α(n)t−O(α(n) t−1) for s ≥ 4 even.
Counting patternfree set partitions. II: Noncrossing and other hypergraphs
 J. Combin
, 2000
"... 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 co ..."
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Cited by 24 (9 self)
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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 StanleyWilf conjecture on permutations. Using generalized DavenportSchinzel 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
Generalized DavenportSchinzel Sequences
, 1993
"... 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 ..."
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Cited by 23 (4 self)
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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).
The number of edges in kquasiplanar graphs
, 2012
"... A graph drawn in the plane is called kquasiplanar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a kquasiplanar graph with n vertices is O(n). The best known upper bound is n(log n) O(log k). In the ..."
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Cited by 18 (3 self)
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A graph drawn in the plane is called kquasiplanar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a kquasiplanar graph with n vertices is O(n). The best known upper bound is n(log n) O(log k). In the present note, we improve this bound to (n log n)2α(n)ck in the special case where the graph is drawn in such a way that every pair of edges meet at most once. Here α(n) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for kquasiplanar graphs in which every edge is drawn as an xmonotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs is at most 2ck6n log n.