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Permutations with short monotone subsequences. (English) Zbl 1109.05015
Summary: We consider permutations of \(1,2,\dots ,n^{2}\) whose longest monotone subsequence is of length \(n\) and are therefore extremal for the Erdős-Szekeres theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square \( n\times n\) 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, and other properties of these extremal permutations.

MSC:
05A05 Permutations, words, matrices
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[1] Edelman, P.H.; Greene, C., Balanced tableaux, Adv. math., 63, 42-99, (1987) · Zbl 0616.05005
[2] Knuth, D.E., The art of computer programming, vol. 3: sorting and searching, (1998), Addison-Wesley
[3] B.G. Pittel, D. Romik, Limit shapes for random square Young tableaux, Adv. in Appl. Math., in press · Zbl 1122.60009
[4] Romik, D., The number of steps in the robinson – schensted algorithm, Funct. anal. appl., 39, 152-155, (2005) · Zbl 1117.05006
[5] Stanley, R.P., Enumerative combinatorics, vol. 2, (1999), Cambridge Univ. Press · Zbl 0928.05001
[6] R.P. Stanley, Private communication, 2005
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