Electronic Journal of Probability

Inversions and longest increasing subsequence for $k$-card-minimum random permutations

Nicholas Travers

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A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider, here, a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower numbered of the remaining cards at each step, and therefore leads to a final permutation which is more ''ordered'' than in the uniform case (i.e. closer to the identity permutation id $=(1,2,3,...,n)$).

We quantify this effect in terms of two natural measures of order: The number of inversions $I$ and the length of the longest increasing subsequence $L$. For inversions, we establish a weak law of large numbers and central limit theorem, both for fixed and growing $k$. For the longest increasing subsequence, we establish the rate of scaling, in general, and existence of a weak law in the case of growing $k$. We also show that the minimum strategy, of selecting the minimum of the $k$ given choices at each step, is optimal for minimizing the number of inversions in the space of all online $k$-card selection rules.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 11, 27 pp.

Accepted: 15 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C
Secondary: 60F

Random Permutation Inversions Longest Increasing Subsequence Power of Choice

This work is licensed under aCreative Commons Attribution 3.0 License.


Travers, Nicholas. Inversions and longest increasing subsequence for $k$-card-minimum random permutations. Electron. J. Probab. 20 (2015), paper no. 11, 27 pp. doi:10.1214/EJP.v20-3602. https://projecteuclid.org/euclid.ejp/1465067117

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