scholarly journals An Erd\H{o}s--Hajnal analogue for permutation classes

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Vincent Vatter
Keyword(s):  
Monotone Subsequence

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.

scholarly journals Monotone Subsequences in High-Dimensional Permutations

2017 ◽  
Vol 27 (1) ◽  
pp. 69-83 ◽  
Author(s):  
NATHAN LINIAL ◽  
MICHAEL SIMKIN
Keyword(s):  
Classical Case ◽  
The Other ◽  
High Dimensional ◽  
Ongoing Effort ◽  
Other Hand ◽  
Monotone Subsequence

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-nk-dimensional permutation contains a monotone subsequence of length Ωk($\sqrt{n}$), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk(nk/(k+1)).

scholarly journals The Minimum Number of Monotone Subsequences

10.37236/1676 ◽  
2002 ◽  
Vol 9 (2) ◽  
Author(s):  
Joseph Samuel Myers
Keyword(s):  
Additional Constraint ◽  
Catalan Number ◽  
Monotone Subsequence ◽  
Minimum Number ◽  
Complete Characterisation

Erdős and Szekeres showed that any permutation of length $n \geq k^2+1$ contains a monotone subsequence of length $k+1$. A simple example shows that there need be no more than $(n \bmod k){{\lceil n/k \rceil}\choose {k+1}} + (k - (n \bmod k)){{\lfloor n/k \rfloor}\choose {k+1}}$ 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 permutations. For $k > 2$ and $n \geq k(2k-1)$, we characterise the permutations containing the minimum number of monotone subsequences of length $k+1$ subject to the additional constraint that all such subsequences go in the same direction (all ascending or all descending); we show that there are $2 {{k}\choose {n \bmod k}} C_k^{2k-2}$ such extremal permutations, where $C_k = {{1}\over {k+1}}{{2k}\choose {k}}$ is the $k^{{\rm th}}$ Catalan number. We conjecture, with some supporting computational evidence, that permutations with a minimum number of monotone $(k+1)$-subsequences must have all such subsequences in the same direction if $n \geq k(2k-1)$, except for the case of $k = 3$ and $n = 16$.

scholarly journals Optimal online selection of a monotone subsequence: a central limit theorem

2015 ◽  
Vol 125 (9) ◽  
pp. 3596-3622 ◽  
Author(s):  
Alessandro Arlotto ◽  
Vinh V. Nguyen ◽  
J. Michael Steele
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Monotone Subsequence ◽  
Selection Of

scholarly journals Growth Rates for Monotone Subsequence

1978 ◽  
Vol 71 (2) ◽  
pp. 179
Author(s):  
A. Del Junco ◽  
J. Michael Steel
Keyword(s):  
Growth Rates ◽  
Monotone Subsequence

scholarly journals Monotone subsequence via ultrapower

2018 ◽  
Vol 16 (1) ◽  
pp. 149-153
Author(s):  
Piotr Błaszczyk ◽  
Vladimir Kanovei ◽  
Mikhail G. Katz ◽  
Tahl Nowik
Keyword(s):  
Infinite Sequence ◽  
Monotone Subsequence

AbstractAn ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and give other applications.

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