2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 5762
author:  Dan Romik 

title:  Permutations with short monotone subsequences 
keywords:  RobinsonSchensted correspondence, ErdősSzekeres theorem, limit shape 
abstract: 
We consider permutations of
1,2,...,n
whose longest monotone subsequence is of length
2
n
and are therefore extremal for the
ErdősSzekeres Theorem. Such permutations
correspond via the RobinsonSchensted correspondence to
pairs of square
n× n
Young tableaux. We show that all the bumping
sequences are constant and therefore these permutations
have a simple description in terms of the pair of square
tableaux. We deduce a limit shape result for the plot of
values of the typical such permutation, which in particular
implies that the first value taken by such a permutation is
with high probability
(1+o(1))n
.
2
/2

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reference:  Dan Romik (2005), Permutations with short monotone subsequences, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 5762 
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