منابع مشابه
Cross-Monotone Subsequences
Two finite real sequences (aI , . . . . ak) and (b, , . . . . bk) are cross-monotone if each is nondecreasing and ai + 1 -ci > bi + 1 bt for all i < k. A sequence ((x, , . . . , on) of nondecreasing reals is in class CM(k) if it has disjoint k-term subsequences that are cross-monotone. The paper shows that f(k), the smallest n such that every nondecreasing (o, , . . . , on) is in CM(k), is boun...
متن کاملMonotone Subsequences in Any Dimension
We exhibit sequences of n points in d dimensions with no long monotone subsequences, by which we mean when projected in a general direction, our sequence has no monotone subsequences of length n+d or more. Previous work proved that this function of n would lie between n and 2 n; this paper establishes that the coefficient of n is one. This resolves the question of how the Erdo s Szekeres result...
متن کاملPermutations with short monotone subsequences
We consider permutations of 1, 2, ..., n whose longest monotone subsequence is of length n and are therefore extremal for the Erdős-Szekeres theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square n × n Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of ...
متن کاملThe Minimum Number of Monotone Subsequences
Erdős and Szekeres showed that any permutation of length n ≥ k2 + 1 contains a monotone subsequence of length k + 1. A simple example shows that there need be no more than (n mod k) (dn/ke k+1 ) + (k − (n mod k))(bn/kc k+1 ) such subsequences; we conjecture that this is the minimum number of such subsequences. We prove this for k = 2, with a complete characterisation of the extremal permutation...
متن کاملMonotone Subsequences in High-Dimensional Permutations
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres Theorem: For every k ≥ 1, every order-n k-dimensional permutation contains a monotone subsequence of length Ωk (√ n ) , and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random kdimensional permutation of order n is...
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ژورنال
عنوان ژورنال: Order
سال: 1985
ISSN: 0167-8094,1572-9273
DOI: 10.1007/bf00582741