The Annals of Applied Probability

The length of the longest common subsequence of two independent mallows permutations

Ke Jin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The Mallows measure is a probability measure on $S_{n}$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q>0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n\to\infty$.

Article information

Ann. Appl. Probab., Volume 29, Number 3 (2019), 1311-1355.

Received: July 2017
First available in Project Euclid: 19 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 05A05: Permutations, words, matrices

Longest common subsequence longest increasing subsequence Mallows measure


Jin, Ke. The length of the longest common subsequence of two independent mallows permutations. Ann. Appl. Probab. 29 (2019), no. 3, 1311--1355. doi:10.1214/17-AAP1351.

Export citation


  • Abello, J. (1991). The weak Bruhat order of $S_{\Sigma}$ consistent sets, and Catalan numbers. SIAM J. Discrete Math. 4 1–16.
  • Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
  • Bhatnagar, N. and Peled, R. (2015). Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields 161 719–780.
  • Chvátal, V. and Sankoff, D. (1975). Longest common subsequences of two random sequences. J. Appl. Probab. 12 306–315.
  • Critchlow, D. E. (1985). Metric Methods for Analyzing Partially Ranked Data. Lecture Notes in Statistics 34. Springer, Berlin.
  • Dančík, V. (1994). Expected length of longest common subsequences. Ph.D. dissertation, Univ. Warwick.
  • Dančík, V. and Paterson, M. (1995). Upper bounds for the expected length of a longest common subsequence of two binary sequences. Random Structures Algorithms 6 449–458.
  • Deken, J. G. (1979). Some limit results for longest common subsequences. Discrete Math. 26 17–31.
  • Deuschel, J.-D. and Zeitouni, O. (1995). Limiting curves for i.i.d. records. Ann. Probab. 23 852–878.
  • Fligner, M. A. and Verducci, J. S. (1993). Probability Models and Statistical Analyses for Ranking Data. Lecture Notes in Statistics 80. Springer, Berlin.
  • Hammersley, J. (1972). A few seedlings of research. In Proc. of the Sixth Berkeley Symp. Math. Statist. and Probability, Vol. 1 345–394. Univ. California Press, Berkeley, CA.
  • Hoppen, C., Kohayakawa, Y., Moreira, C. G., Ráth, B. and Sampaio, R. M. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103 93–113.
  • Houdré, C. and Işlak, Ü. (2014). A central limit theorem for the length of the longest common subsequence in random words. Available at: arXiv:1408.1559.
  • Jin, K. (2017). The limit of the empirical measure of the product of two independent Mallows permutations. Available at: arXiv:1702.00140.
  • Kenyon, R., Kral, D., Radin, C. and Winkler, P. (2015). Permutations with fixed pattern densities. Available at: arXiv:1506.02340.
  • Kerov, S. and Vershik, A. (1977). Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Sov. Math. Dokl. 18 527–531.
  • Lindvall, T. et al. (1999). On Strassen’s theorem on stochastic domination. Electron. Commun. Probab. 4 51–59.
  • Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222.
  • Lueker, G. S. (2009). Improved bounds on the average length of longest common subsequences. J. ACM 56 17.
  • Mallows, C. L. (1957). Non-null ranking models. I. Biometrika 44 114–130.
  • Marden, J. I. (1995). Analyzing and Modeling Rank Data. Monographs on Statistics and Applied Probability 64. Chapman & Hall, London.
  • Mueller, C. and Starr, S. (2013). The length of the longest increasing subsequence of a random Mallows permutation. J. Theoret. Probab. 26 514–540.
  • Mukherjee, S. et al. (2016). Fixed points and cycle structure of random permutations. Electron. J. Probab. 21.
  • Pevzner, P. (2000). Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge, MA.
  • Starr, S. (2009). Thermodynamic limit for the Mallows model on ${S}\_n$. Available at: arXiv:0904.0696.