Monotone Subsequences in Any Dimension
نویسنده
چکیده
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 that a (one-dimensional) sequence has monotone subsequences of at most n generalizes to higher dimensions. 1999 Academic Press
منابع مشابه
The 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 کاملTracking Maximum Ascending Subsequences in Sequences of Partially Ordered Data
We consider scenarios in which long sequences of data are analyzed and subsequences must be traced that are monotone and maximum, according to some measure. A classical example is the online Longest Increasing Subsequence Problem for numeric and alphanumeric data. We extend the problem in two ways: (a) we allow data from any partially ordered set, and (b) we maximize subsequences using much mor...
متن کاملLongest Monotone Subsequences and Rare Regions of Pattern-Avoiding Permutations
We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size n that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or m...
متن کاملOptimal Sequential Selection of a Unimodal Subsequence of a Random Sequence
We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of su...
متن کاملOn Boolean Lattices and Farey Sequences
We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 85 شماره
صفحات -
تاریخ انتشار 1999