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50
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 46 (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 Integer Chebyshev Problem
, 1995
"... . We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail ..."
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Cited by 25 (10 self)
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. We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal "integer Chebyshev" polynomials, showing for example, that on small intevals [0; ffi] and for small degrees d, x d achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys' and others, as to what the "integer transfinite diameter" of [0; 1] should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due t...
Two extensions of Lubinsky’s universality theorem
 J. Anal. Math. 105
, 2008
"... Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1 ..."
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Cited by 20 (9 self)
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Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1
Asymptotic enumeration of permutations avoiding generalized patterns
 Advances in Applied Mathematics 36
, 2006
"... Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permuta ..."
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Cited by 18 (3 self)
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Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns. 1.
Equilibrium measures and capacities in spectral theory
, 2007
"... This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators wh ..."
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Cited by 15 (8 self)
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This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szegő theory.
Not so many runs in strings
, 2008
"... Abstract. Since the work of Kolpakov and Kucherov in [5, 6], it is known that ρ(n), the maximal number of runs in a string, is linear in the length n of the string. A lower bound of 3/(1 + √ 5)n ∼ 0.927n has been given by Franek and al. [3, 4], and upper bounds have been recently provided by Rytter, ..."
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Cited by 12 (1 self)
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Abstract. Since the work of Kolpakov and Kucherov in [5, 6], it is known that ρ(n), the maximal number of runs in a string, is linear in the length n of the string. A lower bound of 3/(1 + √ 5)n ∼ 0.927n has been given by Franek and al. [3, 4], and upper bounds have been recently provided by Rytter, Puglisi and al., and Crochemore and Ilie (1.6n) [8, 7, 1]. However, very few properties are known for the ρ(n)/n function. We show here by a simple argument that limn↦→ ∞ ρ(n)/n exists and that this limit is never reached. Moreover, we further study the asymptotic behavior of ρp(n), the maximal number of runs with period at most p. We provide a new bound for some microruns: we show that there is no more than 0.971n runs of period at most 9 in binary strings. Finally, this technique improves the previous best known upper bound, showing that the total number of runs in a binary string of length n is below 1.52n. 1
On the Minimal Number of Edges in ColorCritical Graphs
, 2001
"... A graph G is kcritical if it has chromatic number k, but every proper subgraph of it is (k 1) colorable. This paper is devoted to investigating the following question: for given k and n, what is the minimal number of edges in a kcritical graph on n vertices, with possibly some additional restrict ..."
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Cited by 9 (3 self)
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A graph G is kcritical if it has chromatic number k, but every proper subgraph of it is (k 1) colorable. This paper is devoted to investigating the following question: for given k and n, what is the minimal number of edges in a kcritical graph on n vertices, with possibly some additional restrictions imposed? Our main result is that for every k 4 and n > k this number is at least k 1 2 + k 3 2(k 2 2k 1) n, thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges in kcritical graphs and provide some constructions of sparse kcritical graphs. A few applications of the results to Ramseytype problems and problems about random graphs are described. 1
The impact of Stieltjes’ work on continued fractions and orthogonal polynomials
, 1993
"... Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials. ..."
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Cited by 9 (0 self)
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Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials.
Small Polynomials With Integer Coefficients
"... this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively wi ..."
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Cited by 8 (6 self)
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this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4], [31], [43], [16], etc.) In the classical case E = [1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) := 2 1n cos(n arccos x), n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so that t n (x) := # b  a 2 # n T n # 2x  a  b b  a # is a monic polynomial with real coe#cients and the smallest uniform norm on [a, b] among all monic polynomials from P n (C). In fact, (1.1) #t n # [a,b] = 2 # b  a 4 # n , n # N, and we find that the Chebyshev constant for [a, b] is given by (1.2) t C ([a, b]) := lim n## #t n # 1/n [a,b] = b  a 4 . The Chebyshev constant of an arbitrary compact set E # C is defined in a similar fashion: (1.3) t C (E) := lim n## #t n # 1/n E , where t n is the Chebyshev polynomial of degree n on E. It is known that t C (E) is equal to the transfinite diameter and the logarithmic capacity cap(E) of the set E (cf. [43, pp. 7175], [16] and [30] for the definitions and background material). 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter, zeros, multiple factors, asymptotic...
Eigenvalues, expanders and gaps between primes
, 2005
"... I kept myself positive, by not getting all negative. ..."
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Cited by 8 (4 self)
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I kept myself positive, by not getting all negative.