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Monotonic subsequences in permutations of n natural numbers


Let Sn be the set of all permutations of the numbers 1, 2,..., n, and letl n(σ) be the number of terms in the maximal monotonic subsequence contained inσ ∈ Sn. If M[l n(σ)] is the mean value ofl n (σ) on Sn, then, for all except a finite number of n, the bound M[l n(σ)] ≤e √n is valid.

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  1. 1.

    Selected Problems and Theorems of Elementary Mathematics [in Russian], Vol. 1, Moscow-Leningrad (1950).

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Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 511–514, April, 1973.

The author wishes to thank E. M. Nikishin for having posed the problem and for his constant interest in the work.

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Stechkin, B.S. Monotonic subsequences in permutations of n natural numbers. Mathematical Notes of the Academy of Sciences of the USSR 13, 310–312 (1973).

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  • Natural Number
  • Finite Number
  • Monotonic Subsequence