TY - JOUR
T1 - Minimum Number of Monotone Subsequences of Length 4 in Permutations
AU - Balogh, József
AU - Hu, Ping
AU - Lidický, Bernard
AU - Pikhurko, Oleg
AU - Udvari, Balázs
AU - Volec, Jan
N1 - Publisher Copyright:
© 2014 Cambridge University Press.
PY - 2015/7/1
Y1 - 2015/7/1
N2 - 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 [EQUATION PRESENTED] 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.
AB - 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 [EQUATION PRESENTED] 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.
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U2 - 10.1017/S0963548314000820
DO - 10.1017/S0963548314000820
M3 - Article
AN - SCOPUS:84939271139
SN - 0963-5483
VL - 24
SP - 658
EP - 679
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 4
ER -