## Electronic Journal of Probability

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

Nicholas Travers

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465067117

Digital Object Identifier
doi:10.1214/EJP.v20-3602

Mathematical Reviews number (MathSciNet)
MR3317153

Zentralblatt MATH identifier
1353.60012

Subjects
Primary: 60C
Secondary: 60F

Rights

#### Citation

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

#### References

• Feller, William. An introduction to probability theory and its applications. Vol. I. Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
• A.M. Vershik and S.V. Kerov. Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. Soviet Math. Dokl., 18:527–531, 1977.
• Logan, B. F.; Shepp, L. A. A variational problem for random Young tableaux. Advances in Math. 26 (1977), no. 2, 206–222.
• Aldous, D.; Diaconis, P. Hammersley's interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 (1995), no. 2, 199–213.
• Baik, Jinho; Deift, Percy; Johansson, Kurt. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178.
• Azar, Yossi; Broder, Andrei Z.; Karlin, Anna R.; Upfal, Eli. Balanced allocations. SIAM J. Comput. 29 (1999), no. 1, 180–200.
• Bohman, Tom; Kravitz, David. Creating a giant component. Combin. Probab. Comput. 15 (2006), no. 4, 489–511.
• Spencer, Joel; Wormald, Nicholas. Birth control for giants. Combinatorica 27 (2007), no. 5, 587–628.
• Krivelevich, Michael; Loh, Po-Shen; Sudakov, Benny. Avoiding small subgraphs in Achlioptas processes. Random Structures Algorithms 34 (2009), no. 1, 165–195.
• Krivelevich, Michael; Spóhel, Reto. Creating small subgraphs in Achlioptas processes with growing parameter. SIAM J. Discrete Math. 26 (2012), no. 2, 670–686.
• Mallows, C. L. Non-null ranking models. I. Biometrika 44 (1957), 114–130.
• N. Bhatnagar and R. Peled. Lengths of monotone subsequences in a Mallows permutation. arxiv.org 1306.3674, 2013.
• Rabinovitch, Peter. Uniform and Mallows Random Permutations: Inversions, Levels & Sampling. Thesis (Ph.D.)-Carleton University (Canada). ProQuest LLC, Ann Arbor, MI, 2012. 91 pp. ISBN: 978-0494-89317-3
• Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8
• Alon, Noga; Spencer, Joel H. The probabilistic method. Second edition. With an appendix on the life and work of Paul ErdÅ's. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, 2000. xviii+301 pp. ISBN: 0-471-37046-0
• S. Boucheron, G. Lugosi, and O. Bousquet. Concentration inequalities. Lecture Notes in Computer Science, 3176:208–240, 2004.
• Steele, J. Michael. An Efron-Stein inequality for nonsymmetric statistics. Ann. Statist. 14 (1986), no. 2, 753–758.