Skip to main content

Bounded Affine Permutations II. Avoidance of Decreasing Patterns

Abstract

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size N that avoid the monotone decreasing pattern of fixed size m. We prove that the number of such permutations is asymptotically equal to \((m-1)^{2N} N^{(m-2)/2}\) times an explicit constant as \(N\rightarrow \infty \). For instance, the number of bounded affine permutations of size N that avoid 321 is asymptotically equal to \(4^N (N/4\pi )^{1/2}\). We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding \(m\cdots 1\) looks like \(m-1\) random lines of slope 1 whose y intercepts sum to 0.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Availability of data and material

Not applicable.

Notes

  1. 1.

    Proportional to one-dimensional Lebesgue measure.

References

  1. 1.

    D. André, Mémoire sur les combinaisons régulières et leurs applications, Ann. Sci. Éc. Norm. Supér. (2) 5 (1876), 155–198.

    Article  Google Scholar 

  2. 2.

    R. Arratia, On the Stanley–Wilf conjecture for the number of permutations avoiding a given pattern, Electron. J. Combin. 6 (1999), no. 1, N1.

  3. 3.

    F. Bassino, M. Bouvel, V. Féray, L. Gerin, M. Maazoun, and A. Pierrot, Universal limits of substitution-closed permutation classes, J. Eur. Math. Soc. 22 (2020), 3565–3639.

    MathSciNet  Article  Google Scholar 

  4. 4.

    D. Bevan, Permutation patterns: basic definitions and notation, arXiv:1506.06673 [math.CO] (2015).

  5. 5.

    S. Billey and A. Crites, Pattern characterization of rationally smooth affine Schubert varieties of type \(A\), J. Algebra 361 (2012), 107–133.

    MathSciNet  Article  Google Scholar 

  6. 6.

    A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math. 231, Springer, New York, 2005.

    MATH  Google Scholar 

  7. 7.

    M. Bóna, Combinatorics of Permutations, Chapman and Hall/CRC, Boca Raton, 2004.

    Book  Google Scholar 

  8. 8.

    Miklós Bóna, The absence of a pattern and the occurrences of another, Discrete Math. Theor. Comput. Sci. 12 (2010), 89–102.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    J. Borga, M. Bouvel, V. Féray, and B. Stufler, A decorated tree approach to random permutations in substitution-closed classes, Electron. J. Probab. 25 (2020), 1–52.

    MathSciNet  Article  Google Scholar 

  10. 10.

    J. Borga and M. Maazoun, Scaling and local limits of Baxter permutations through coalescent-walk process, arXiv:2003.09086 [math.PR] (2020).

  11. 11.

    F. Borga and E. Slivken, Square permutations are typically rectangular, Ann. Appl. Probab. 30 (2020), 2196–2233.

    MathSciNet  Article  Google Scholar 

  12. 12.

    C. C. S. Caiado and P. N. Rathie, Polynomial coefficients and distribution of the sum of discrete uniform variables, Eighth Annual Conference of the Society for Special Functions and Their Applications, Pala, India, Society for Special Functions and Their Applications, 2007.

  13. 13.

    M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, World Scientific, Singapore, 1992.

    Book  Google Scholar 

  14. 14.

    S.-E. Cheng, S.-P. Eu, and T.-S. Fu, Area of Catalan paths on a checkerboard, European J. Combin. 28 (2007), 1331–1344.

    MathSciNet  Article  Google Scholar 

  15. 15.

    N. Clisby, Endless self-avoiding walks, J. Phys. A: Math. Theor. 46 (2013), 235001, 32 pp.

  16. 16.

    A. Crites, Enumerating pattern avoidance for affine permutations, Electron. J. Combin. 17 (2010), #R127.

  17. 17.

    S. Elizalde, Fixed points and excedances in restricted permutations, Electron. J. Combin. 18 (2) (2012), #P29.

  18. 18.

    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I (Third Edition), Wiley, New York, 1968.

    MATH  Google Scholar 

  19. 19.

    R. Glebov, A. Grzesik, T. Klimošová, and D. Král, Finitely forcible graphons and permutons, J. Combin. Theor. Ser. B 110 (2015), 112–135.

    MathSciNet  Article  Google Scholar 

  20. 20.

    W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.

    MathSciNet  Article  Google Scholar 

  21. 21.

    C. Hoppen, Y. Kohoyakawa, C. G. Moreira, B. Ráth, and R. M. Sampaio, Limits of permutation sequences, J. Comb. Theory B 103 (2013), 93–113.

    MathSciNet  Article  Google Scholar 

  22. 22.

    S. Janson, Patterns in random permutations avoiding the pattern \(132\), Combin. Probab. Comput. 26 (2017), 24–51.

    MathSciNet  Article  Google Scholar 

  23. 23.

    S. Janson, Patterns in random permutations avoiding the pattern \(321\), Random Structures Algorithms 55 (2019), 249–270.

    MathSciNet  Article  Google Scholar 

  24. 24.

    N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, Second Edition, Wiley, New York, 1995.

  25. 25.

    R. Kenyon, D. Král’, C. Radin, and P. Winkler, Permutations with fixed pattern densities, Random Struct. Alg. 56 (2020), 220–250. https://doi.org/10.1002/rsa.20882

  26. 26.

    A. Knutson, T. Lam, and D. E. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), 1710–1752.

    MathSciNet  Article  Google Scholar 

  27. 27.

    D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, Providence, 2009.

    MATH  Google Scholar 

  28. 28.

    N. Madras and L. Pehlivan, Large deviations for permutations avoiding monotone patterns, Electron. J. Combin. 23 (2016), #P4.36.

  29. 29.

    N. Madras and J. M. Troyka, Bounded affine permutations I. Pattern avoidance and enumeration, Discrete Math. Theor. Comput. Sci. 22 (2) (2021), #1.

  30. 30.

    A. Marcus and G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture, J. Combin. Theor. Ser. A 107 (2004), 153–160.

    MathSciNet  Article  Google Scholar 

  31. 31.

    C. B. Presutti and W. R. Stromquist, Packing rates of measures and a conjecture for the packing density of 2413, Lond. Math. Soc. Lecture Notes 376 (2010), 3–40.

    MathSciNet  MATH  Google Scholar 

  32. 32.

    A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. Math. 41 (1981), 115–136.

    MathSciNet  Article  Google Scholar 

  33. 33.

    L. B. Richmond and J. Shallit, Counting abelian squares, Electron. J. Combin. 16 (2009), #R72.

  34. 34.

    V. Vatter, Permutation classes, Handbook of Enumerative Combinatorics, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2015.

Download references

Acknowledgements

We are grateful to Tom Salisbury for a helpful discussion about random measures, and to the anonymous referees for helpful comments.

Funding

The research was pursued as part of a Discovery Grant to N. Madras from NSERC Canada.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Neal Madras.

Ethics declarations

Conflict of interest

None.

Code availability

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

N. Madras was supported in part by a Discovery Grant from NSERC Canada, and by a Minor Research Grant from the Faculty of Science at York University.

Communicated by David Bevan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Madras, N., Troyka, J.M. Bounded Affine Permutations II. Avoidance of Decreasing Patterns. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00553-4

Download citation

Keywords

  • Permutation
  • Affine permutation
  • Permutation pattern
  • Asymptotic enumeration
  • Permuton
  • Random measure

Mathematics Subject Classification

  • Primary 05A05
  • 05A16
  • 60C05
  • 60G57