Geodesic
Monotonic subsequences in permutations of $n$ natural numbers
Matematičeskie zametki, Tome 13 (1973) no. 4, pp. 511-514.

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Let $S_n$ be the set of all permutations of the numbers $1,2,\dots,n$, and let $l_n(\sigma)$ be the number of terms in the maximal monotonic subsequence contained in $\sigma\in S_n$. If $M(l_n(\sigma))$ is the mean value of $l_n(\sigma)$ on $S_n$, then, for all except a finite number of n, the bound $M(l_n(\sigma))\le e\sqrt n$ is valid. 
@article{MZM_1973_13_4_a3,
     author = {B. S. Stechkin},
     title = {Monotonic subsequences in permutations of $n$ natural numbers},
     journal = {Matemati\v{c}eskie zametki},
     pages = {511--514},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_4_a3/}
}
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B. S. Stechkin. Monotonic subsequences in permutations of $n$ natural numbers. Matematičeskie zametki, Tome 13 (1973) no. 4, pp. 511-514. http://geodesic.mathdoc.fr/item/MZM_1973_13_4_a3/