The Annals of Applied Probability

The Height of a Random Partial Order: Concentration of Measure

Bela Bollobas and Graham Brightwell

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The problem of determining the length $L_n$ of the longest increasing subsequence in a random permutation of $\{1, \ldots, n\}$ is equivalent to that of finding the height of a random two-dimensional partial order (obtained by intersecting two random linear orders). The expectation of $L_n$ is known to be about $2\sqrt{n}$. Frieze investigated the concentration of $L_n$ about this mean, showing that, for $\varepsilon > 0$, there is some constant $\beta > 0$ such that $Pr(|L_n - \mathbf{E}L_n| \geq n^{1/3+\varepsilon}) \leq \exp(-n^\beta).$ In this paper we obtain similar concentration results for the heights of random $k$-dimensional orders, for all $k \geq 2$. In the case $k = 2$, our method replaces the $n^{1/3+\varepsilon}$ above with $n^{1/4+\varepsilon}$, which we believe to be essentially best possible.

Article information

Ann. Appl. Probab., Volume 2, Number 4 (1992), 1009-1018.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 06A10
Secondary: 60C05: Combinatorial probability 05A99: None of the above, but in this section

Partial order height random orders Ulam's problem increasing subsequences


Bollobas, Bela; Brightwell, Graham. The Height of a Random Partial Order: Concentration of Measure. Ann. Appl. Probab. 2 (1992), no. 4, 1009--1018. doi:10.1214/aoap/1177005586.

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