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Patterns in Random Permutations Avoiding the Pattern 132

  • SVANTE JANSON (a1)
Abstract

We consider a random permutation drawn from the set of 132-avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by n λ(σ)/2, where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

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[1] Albert, M. H., Atkinson, M. D. and Brignall, R. (2011) The enumeration of permutations avoiding 2143 and 4231. Pure Math. Appl. 22 8798.
[2] Albert, M. H., Atkinson, M. D. and Brignall, R. (2012) The enumeration of three pattern classes using monotone grid classes. Electron. J. Combin. 19 20.
[3] Aldous, D. (1991) The continuum random tree II: An overview. In Stochastic Analysis: Durham 1990, Vol. 167 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 2370.
[4] Aldous, D. (1993) The continuum random tree III. Ann. Probab. 21 248289.
[5] Biane, P., Pitman, J. and Yor, M. (2001) Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 435465.
[6] Billey, S. C., Jockusch, W. and Stanley, R. P. (1993) Some combinatorial properties of Schubert polynomials. J. Algebraic Combin. 2 345374.
[7] Billingsley, P. (1968) Convergence of Probability Measures, Wiley.
[8] Bóna, M. (2004) Combinatorics of Permutations, Chapman & Hall/CRC.
[9] Bóna, M. (2007) The copies of any permutation pattern are asymptotically normal. arXiv:0712.2792
[10] Bóna, M. (2010) The absence of a pattern and the occurrences of another. Discrete Math. Theor. Comput. Sci. 12 89102.
[11] Bóna, M. (2010) On three different notions of monotone subsequences. In Permutation Patterns, Vol. 376 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 89114.
[12] Bóna, M. (2012) Surprising symmetries in objects counted by Catalan numbers. Electron. J. Combin. 19 62.
[13] Bousquet-Mélou, M. and Janson, S. (2006) The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 15971632.
[14] Cheng, S.-E., Eu, S.-P. and Fu, T.-S. (2007) Area of Catalan paths on a checkerboard. European J. Combin. 28 13311344.
[15] Chow, T. and West, J. (1999) Forbidden subsequences and Chebyshev polynomials. Discrete Math. 204 119128.
[16] Cooper, J. Combinatorial problems I like. http://www.math.sc.edu/~cooper/combprob.html
[17] Drmota, M. (2009) Random Trees, Springer.
[18] Durrett, R. T., Iglehart, D. L. and Miller, D. R. (1977) Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5 117129.
[19] Fill, J. A. and Janson, S. (2009) Precise logarithmic asymptotics for the right tails of some limit random variables for random trees. Ann. Combin. 12 403416.
[20] Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.
[21] Gut, A. (2013) Probability: A Graduate Course, second edition, Springer.
[22] Homberger, C. (2012) Expected patterns in permutation classes. Electron. J. Combin. 19 43.
[23] Janson, S. (2003) The Wiener index of simply generated random trees. Random Struct. Alg. 22 337358.
[24] Janson, S. (2007) Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 80145.
[25] Janson, S. Patterns in random permutations avoiding the pattern 132. (Earlier version of the present paper.) arXiv:1401.5679v1
[26] Janson, S., Nakamura, B. and Zeilberger, D. (2015) On the asymptotic statistics of the number of occurrences of multiple permutation patterns. J. Combin. 6 117143.
[27] Kallenberg, O. (2002) Foundations of Modern Probability, second edition, Springer.
[28] Knuth, D. E. (1997) The Art of Computer Programming, Vol. 1: Fundamental Algorithms, third edition, Addison-Wesley.
[29] Krattenthaler, C. (2001) Permutations with restricted patterns and Dyck paths. Adv. Appl. Math. 27 510530.
[30] Louchard, G. (1984) The Brownian excursion area: A numerical analysis. Comput. Math. Appl. 10 413417. Erratum: Comput. Math. Appl. A 12 (1986) 375.
[31] Mansour, T. and Vainshtein, A. (2000) Restricted permutations, continued fractions, and Chebyshev polynomials. Electron. J. Combin. 7 R17.
[32] Mansour, T. and Vainshtein, A. (2001) Restricted 132-avoiding permutations. Adv. Appl. Math. 26 258269.
[33] Mansour, T. and Vainshtein, A. (2001/02) Restricted permutations and Chebyshev polynomials. Sém. Lothar. Combin. 47 B47c.
[34] Marckert, J.-F. (2004) The rotation correspondence is asymptotically a dilatation. Random Struct. Alg. 24 118132.
[35] Marckert, J.-F. and Mokkadem, A. (2003) The depth first processes of Galton–Watson trees converge to the same Brownian excursion. Ann. Probab. 31 16551678.
[36] Nguyen The, M. (2004) Area and inertial moment of Dyck paths. Combin. Probab. Comput. 13 697716.
[37] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, third edition, Springer.
[38] Richard, C. (2009) On q-functional equations and excursion moments. Discrete Math. 309 207230.
[39] Robertson, A., Wilf, H. S. and Zeilberger, D. (1999) Permutation patterns and continued fractions. Electron. J. Combin. 6 R38.
[40] Rudolph, K. (2013) Pattern popularity in 132-avoiding permutations. Electron. J. Combin. 20 8.
[41] Simion, R. and Schmidt, F. W. (1985) Restricted permutations. European J. Combin. 6 383406.
[42] Stanley, R. P. (1999) Enumerative Combinatorics, Vol. 2, Cambridge University Press.
[43] West, J. (1996) Generating trees and forbidden subsequences. Discrete Math. 157 363374.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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