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### Minimum Number of Monotone Subsequences of Length 4 in Permutations

JÓZSEF BALOGH, PING HU, BERNARD LIDICKÝ, OLEG PIKHURKO, BALÁZS UDVARI, JAN VOLEC
2014 Combinatorics, probability & computing
We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n  ...  This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.  ...  Acknowledgement We would like to thank Dan Krá , l for fruitful discussions and developing the first version of the program implementing flag algebras over permutations.  ...

### The Minimum Number of Monotone Subsequences

Joseph Samuel Myers
2002 Electronic Journal of Combinatorics
Erdős and Szekeres showed that any permutation of length $n \geq k^2+1$ contains a monotone subsequence of length $k+1$.  ...  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  ...  We write m k (S) for the number of monotone subsequences of length k + 1 in the permutation S.  ...

### Page 7609 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews
7609 2004j:05008 05A05 05D99 Myers, Joseph Samuel (4-CAMB-CM; Cambridge) The minimum number of monotone subsequences. (English summary) Permutation patterns (Otago, 2003). Electron. J.  ...  There is also some computa- tional evidence to support the conjecture that all permutations of length » with n > k(2k —1) and a minimum number of monotone subsequences of length A +1, must have all such  ...

### On minimum k-modal partitions of permutations

Gabriele Di Stefano, Stefan Krause, Marco E. Lübbecke, Uwe T. Zimmermann
2008 Journal of Discrete Algorithms
Partitioning a permutation into a minimum number of monotone subsequences is N P-hard.  ...  For the online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze  ...  In particular for 1-modal, or unimodal, subsequences Chung [5] proves that any permutation of length n contains such a subsequence of length √ 3(n − 1/4) − 1/2 , and this is best possible.  ...

### On Minimum k-Modal Partitions of Permutations [chapter]

Gabriele Di Stefano, Stefan Krause, Marco E. Lübbecke, Uwe T. Zimmermann
2006 Lecture Notes in Computer Science
Partitioning a permutation into a minimum number of monotone subsequences is N P-hard.  ...  For the online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze  ...  In particular for 1-modal, or unimodal, subsequences Chung [5] proves that any permutation of length n contains such a subsequence of length √ 3(n − 1/4) − 1/2 , and this is best possible.  ...

### Crucial and bicrucial permutations with respect to arithmetic monotone patterns [article]

Sergey Avgustinovich, Sergey Kitaev, Alexandr Valyuzhenich
2012 arXiv   pre-print
A pattern τ is a permutation, and an arithmetic occurrence of τ in (another) permutation π=π_1π_2...π_n is a subsequence π_i_1π_i_2...π_i_m of π that is order isomorphic to τ where the numbers i_1<i_2<  ...  Moreover, we show that the minimal length of a (k,ℓ)-crucial permutation is (k,ℓ)((k,ℓ)-1), while the minimal length of a (k,ℓ)-bicrucial permutation is at most 2(k,ℓ)((k,ℓ)-1), again for k,ℓ≥3.  ...  monotone subsequences of length 4 or more can have at most one large white element and one small white element leading to the fact that a monotone subsequence of the permutation in Figure 6 are of length  ...

### An Erdős--Hajnal analogue for permutation classes [article]

Vincent Vatter
2016 arXiv   pre-print
We prove that there is a constant c such that every permutation in C of length n contains a monotone subsequence of length cn.  ...  Let C be a permutation class that does not contain all layered permutations or all colayered permutations.  ...  There is a constant c > 0 such that every permutation of length n in C contains a monotone subsequence of length at least cn. One special case of Theorem 1 is quite easy.  ...

### Avoidance of boxed mesh patterns on permutations

Sergey Avgustinovich, Sergey Kitaev, Alexandr Valyuzhenich
2013 Discrete Applied Mathematics
Finally, we discuss enumeration of permutations avoiding simultaneously two or more length-three boxed mesh patterns, where we meet generalized Catalan numbers.  ...  We introduce the notion of a boxed mesh pattern and study avoidance of these patterns on permutations.  ...  The third author was also supported by the grant of the President of the Russian Federation for Young Russian researchers (project no. MK-4075.2012.1)  ...

### A quadratic time 2-approximation algorithm for block sorting

Wolfgang W. Bein, Lawrence L. Larmore, Linda Morales, I. Hal Sudborough
2009 Theoretical Computer Science
The block sorting problem is the problem of minimizing the number of steps to sort a list of distinct items, where a sublist of items which are already in sorted order, called a block, can be moved in  ...  Block sorting has importance in connection with optical character recognition (OCR) and is related to transposition sorting in computational biology.  ...  This author worked on this project at the University of Texas at Dallas while on sabbatical leave from UNLV. Second author is supported by NSF grant CCR-0312093.  ...

### On compressing permutations and adaptive sorting

Jérémy Barbay, Gonzalo Navarro
2013 Theoretical Computer Science
We prove that, given a permutation π over [1..n] formed of nRuns sorted blocks of sizes given by the vector R = r 1 , . . . , r nRuns , there exists a compressed data structure encoding n r i n(1 + log  ...  2 nRuns) bits while supporting access to the values of π () and π −1 () in time O(log nRuns/ log log n) in the worst case and O(H(R)/ log log n) on average, when the argument is uniformly distributed  ...  on them based on optimality implications in terms of the number of comparisons performed.  ...

### Generalised Pattern Avoidance [article]

Anders Claesson
2000 arXiv   pre-print
We consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters.  ...  We also give some results for the number of permutations avoiding two different patterns.  ...  Acknowledgement I am greatly indebted to my advisor Einar Steingrímsson, who put his trust in me and gave me the opportunity to study mathematics on a postgraduate level.  ...

### Two RPG Flow-graphs for Software Watermarking using Bitonic Sequences of Self-inverting Permutations [article]

Anna Mpanti, Stavros D. Nikolopoulos
2016 arXiv   pre-print
Following up on our recently proposed methods for encoding watermark numbers w as reducible permutation flow-graphs F[π^*] through the use of self-inverting permutations π^*, in this paper, we extend the  ...  bitonic subsequences composing the self-inverting permutation π^*.  ...  In this paper, we consider only bitonic sequences that monotonically increases and then monotonically decreases, i.e., the minimum element of such a sequence b is either the first b 1 or the last b n element  ...

### Upper Bound Constructions for Untangling Planar Geometric Graphs

Javier Cano, Csaba D. Tóth, Jorge Urrutia
2014 SIAM Journal on Discrete Mathematics
as in Dn.  ...  For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in any crossing-free straight-line drawing of the graph, at most O(n .4982 ) vertices lie at the same position  ...  By Lemma 1, the spread of the monotone subsequence of length at least i /4 is at least ( 2 i + 32)/96. Hence these fixed points "occupy" an interval of length ( 2 i + 32)/96 on the x-axis.  ...

### Upper Bound Constructions for Untangling Planar Geometric Graphs [chapter]

Javier Cano, Csaba D. Tóth, Jorge Urrutia
2012 Lecture Notes in Computer Science
as in Dn.  ...  For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in any crossing-free straight-line drawing of the graph, at most O(n .4982 ) vertices lie at the same position  ...  By Lemma 1, the spread of the monotone subsequence of length at least i /4 is at least ( 2 i + 32)/96. Hence these fixed points "occupy" an interval of length ( 2 i + 32)/96 on the x-axis.  ...

### Generalized Pattern Avoidance

Anders Claesson
2001 European journal of combinatorics (Print)
We will consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters.  ...  We also give some results for the number of permutations avoiding two different patterns.  ...  Acknowledgement I am greatly indebted to my advisor Einar Steingrímsson, who put his trust in me and gave me the opportunity to study mathematics on a postgraduate level.  ...
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