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Minimum Number of Monotone Subsequences of Length 4 in Permutations
2014
Combinatorics, probability & computing
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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. ...
doi:10.1017/s0963548314000820
fatcat:6rh274ecujdhffngmkly34z22u
The Minimum Number of Monotone Subsequences
2002
Electronic Journal of Combinatorics
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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. ...
doi:10.37236/1676
fatcat:y3rl5xf2kzdeffrk2lbzbqwf7m
Page 7609 of Mathematical Reviews Vol. , Issue 2004j
[page]
2004
Mathematical Reviews
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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
2008
Journal of Discrete Algorithms
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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. ...
doi:10.1016/j.jda.2008.01.002
fatcat:xpik3ui5znfxbcux2tje2pmd3m
On Minimum k-Modal Partitions of Permutations
[chapter]
2006
Lecture Notes in Computer Science
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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. ...
doi:10.1007/11682462_36
fatcat:yi3v5dw6mzgyji2nw5echeqb3u
Crucial and bicrucial permutations with respect to arithmetic monotone patterns
[article]
2012
arXiv
pre-print
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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 ...
arXiv:1210.2621v1
fatcat:qmkjgj54nrditcqjfevd3fzgdq
An Erdős--Hajnal analogue for permutation classes
[article]
2016
arXiv
pre-print
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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. ...
arXiv:1511.01076v2
fatcat:zvqtnykvfnbkbk5ihs4euuabae
Avoidance of boxed mesh patterns on permutations
2013
Discrete Applied Mathematics
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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) ...
doi:10.1016/j.dam.2012.08.015
fatcat:gsdyihucfzetnky4trw647nkfm
A quadratic time 2-approximation algorithm for block sorting
2009
Theoretical Computer Science
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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. ...
doi:10.1016/j.tcs.2008.10.022
fatcat:zho2rjgxireghodcuy6mskg4oe
On compressing permutations and adaptive sorting
2013
Theoretical Computer Science
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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. ...
doi:10.1016/j.tcs.2013.10.019
fatcat:vxgaveeqkbhtdeuhbtfcczeery
Generalised Pattern Avoidance
[article]
2000
arXiv
pre-print
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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. ...
arXiv:math/0011235v1
fatcat:zilhpaytb5ebbplz47ogoywglu
Two RPG Flow-graphs for Software Watermarking using Bitonic Sequences of Self-inverting Permutations
[article]
2016
arXiv
pre-print
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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 ...
arXiv:1607.02281v2
fatcat:ljefcwmiozb7ppjbsr3gndcxwy
Upper Bound Constructions for Untangling Planar Geometric Graphs
2014
SIAM Journal on Discrete Mathematics
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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. ...
doi:10.1137/130924172
fatcat:mvpohhbiyrbmpcxzlkutnzzfie
Upper Bound Constructions for Untangling Planar Geometric Graphs
[chapter]
2012
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-642-25878-7_28
fatcat:l3tn3jiufvfxndwtcxlrhkocj4
Generalized Pattern Avoidance
2001
European journal of combinatorics (Print)
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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. ...
doi:10.1006/eujc.2001.0515
fatcat:q43shpidzfftdaqlkgd2nldqeu
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