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Monotone Subsequences in High-Dimensional Permutations

Part of: Designs and configurations

Published online by Cambridge University Press:  16 October 2017

NATHAN LINIAL  and
MICHAEL SIMKIN
NATHAN LINIAL
Affiliation:
School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: [email protected])
MICHAEL SIMKIN
Affiliation:
Institute of Mathematics and Federmann Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: [email protected])
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Abstract

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk($\sqrt{n}$), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk(nk/(k+1)).

MSC classification

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 1 , January 2018 , pp. 69 - 83
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Baik, J., Deift, P. and Johansson, K. (1999) On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.Google Scholar
[2] Baker, R. C., Harman, G. and Pintz, J. (2001) The difference between consecutive primes II. Proc. London Math. Soc. 83 532–562.CrossRefGoogle Scholar
[3] Bollobás, B. and Winkler, P. (1988) The longest chain among random points in Euclidean space. In Proc. Amer. Math. Soc. 103 347–353.CrossRefGoogle Scholar
[4] Dilworth, R. P. (1950) A decomposition theorem for partially ordered sets. Ann. of Math. 51 161–166.CrossRefGoogle Scholar
[5] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2 463–470.Google Scholar
[6] Hammersley, J. (1972) A few seedlings of research. In Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 345–394.Google Scholar
[7] Impagliazzo, R. and Kabanets, V. (2010) Constructive proofs of concentration bounds. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (Serna, M. et al., eds), Springer, pp. 617–631.Google Scholar
[8] Jacobson, M. T. and Matthews, P. (1996) Generating uniformly distributed random Latin squares. J. Combin. Designs 4 405–437.Google Scholar
[9] Kruskal, J. B. (1953) Monotonic subsequences. Proc. Amer. Math. Soc. 4 264–274.Google Scholar
[10] Linial, N. and Luria, Z. (2014) On the vertices of the d-dimensional Birkhoff polytope. Discrete Comput. Geom. 51 161–170.Google Scholar
[11] Linial, N. and Luria, Z. (2014) An upper bound on the number of high-dimensional permutations. Combinatorica 34 471–486.Google Scholar
[12] Logan, B. F. and Shepp, L. A. (1977) A variational problem for random Young tableaux. Adv. Math. 26 206–222.CrossRefGoogle Scholar
[13] Mirsky, L. (1971) A dual of Dilworth's decomposition theorem. Amer. Math. Monthly 78 876–877.CrossRefGoogle Scholar
[14] Steele, J. M. (1995) Variations on the monotone subsequence theme of Erdős and Szekeres. In Discrete Probability and Algorithms (Aldous, D. et al., eds), Springer, pp. 111–131.CrossRefGoogle Scholar
[15] Szabó, T. and Tardos, G. (2001) A multidimensional generalization of the Erdős–Szekeres lemma on monotone subsequences. Combin. Probab. Comput. 10 557–565.Google Scholar
[16] Ulam, S. M. (1961) Monte Carlo calculations in problems of mathematical physics. In Modern Mathematics for the Engineer: Second Series (Beckenbach, E. F., ed.), McGraw-Hill, pp. 261–281.Google Scholar
[17] Vershik, A. M. and Kerov, S. V. (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables. Doklady Akademii Nauk SSSR 233 1024–1027.Google Scholar