Monotonic subsequences in permutations of n natural numbers

Let S_n be the set of all permutations of the numbers 1, 2,..., n, and letl _n() be the number of terms in the maximal monotonic subsequence contained in S_n. If Ml _n()] is the mean value ofl _n () on S_n, then, for all except a finite number of n, the bound Ml _n()] e n is valid.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.