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Minimum Number of Monotone Subsequences of Length 4 in Permutations

Part of: Combinatorics Graph theory Extremal combinatorics

Published online by Cambridge University Press:  19 December 2014

JÓZSEF BALOGH ,
PING HU ,
BERNARD LIDICKÝ ,
OLEG PIKHURKO ,
BALÁZS UDVARI  and
JAN VOLEC
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu) Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail: udvarib@math.u-szeged.hu)
PING HU
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu)
BERNARD LIDICKÝ
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu, lidicky@illinois.edu) Charles University, Prague, Czech Republic
OLEG PIKHURKO
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail: O.Pikhurko@warwick.ac.uk, honza@ucw.cz)
BALÁZS UDVARI
Affiliation:
Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail: udvarib@math.u-szeged.hu)
JAN VOLEC
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail: O.Pikhurko@warwick.ac.uk, honza@ucw.cz)
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Abstract

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*}\binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}.\end{equation*}
Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromatic K4 is minimized. We show that all the extremal colourings must contain monochromatic K4 only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D99: None of the above, but in this section 05A05: Permutations, words, matrices

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 4: Oberwolfach Special Issue Part 1 , July 2015 , pp. 658 - 679
Copyright
Copyright © Cambridge University Press 2014 

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