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Minimum Number of Monotone Subsequences of Length 4 in Permutations

Published online by Cambridge University Press:  19 December 2014

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:,, Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:,,
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:,, Charles University, Prague, Czech Republic
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail:,
Bolyai Institute, University of Szeged, Szeged, Hungary (e-mail:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail:,


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.

Copyright © Cambridge University Press 2014 

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