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Central Limit Theorem for Peaks of a Random Permutation in a Fixed Conjugacy Class of \(S_n\)


The number of peaks of a random permutation is known to be asymptotically normal. We give a new proof of this and prove a central limit theorem for the distribution of peaks in a fixed conjugacy class of the symmetric group. Our technique is to apply analytic combinatorics to study a complicated but exact generating function for peaks in a given conjugacy class.

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  1. Bayer, D. and Diaconis, P., Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2 (1992), 294–313.

    MathSciNet  Article  Google Scholar 

  2. Billey, S., Burdzy, K. and Sagan, B., Permutations with given peak set, J. Integer Seq. 16 (2013), Article 13.6.1.

  3. David, F. and Barton, D., Combinatorial chance, Hafner Publishing Co., 1962.

  4. Diaconis, P., Group representations in probability and statistics, Institute of Mathematical Statistics, Hayward, CA, 1988.

    MATH  Google Scholar 

  5. Diaconis, P., Fulman, J. and Holmes, S., Analysis of casino shelf shuffling machines, Annals Appl. Probab. 23 (2013), 1692–1720.

    MathSciNet  Article  Google Scholar 

  6. Diaconis, P. and Graham, R., Magical mathematics. The mathematical ideas that animate great magic tricks, Princeton University Press, 2012.

  7. Diaconis, P., McGrath, M. and Pitman, J., Riffle shuffles, cycles, and descents, Combinatorica 15 (1995), 11–29.

    MathSciNet  Article  Google Scholar 

  8. Fulman, J., Stein’s method and non-reversible Markov chains, in: Stein’s method: expository lectures and applications, 69-77, IMS Lecture Notes Monogr. Ser., 46, Inst. Math. Statist., 2004.

  9. Fulman, J., The distribution of descents in fixed conjugacy classes of the symmetric groups, J. Combin. Theory Ser. A 84 (1998), 171–180.

    MathSciNet  Article  Google Scholar 

  10. Fulman, J., Neumann, P. and Praeger, C., A generating function approach to the enumeration of matrices in classical groups over finite fields, Mem. Amer. Math. Soc. 176 (2005), no. 830.

  11. Gessel, I. and Reutenauer, C., Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993), 189–215.

    MathSciNet  Article  Google Scholar 

  12. Graham, R.L., Knuth, D.E., Patashnik, O., Concrete mathematics: a foundation for computer science, 2nd ed. Addison-Wesley, Reading, Mass. (1994).

  13. Harper, L., Stirling behavior is asymptotically normal, Ann. Math. Stat. 38 (1966), 410–414.

    MathSciNet  Article  Google Scholar 

  14. Kim, G., Distribution of descents in matchings, Annals Combin. 23 (2019), 73–87.

    MathSciNet  Article  Google Scholar 

  15. Kim, G. and Lee, S., Central limit theorems for descents in conjugacy classes of \(S_n\), J. Combin. Theory Ser. A 169 (2020), 105123,

    MathSciNet  Article  Google Scholar 

  16. Knuth, D., The art of computer programming, Volume 3. Sorting and searching, Addison-Wesley, 1973.

  17. Nyman, K., The peak algebra of the symmetric group, J. Algebraic Combin. 17 (2003), 309–322.

    MathSciNet  Article  Google Scholar 

  18. Petersen, K., Eulerian numbers, Birkhauser, 2015.

  19. Petersen, K., Enriched \(P\)-partitions and peak algebras, Adv. Math. 209 (2007), 561–610.

    MathSciNet  Article  Google Scholar 

  20. Pitman, J., Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Combin. Theory Ser. A 77 (1997), 279–303.

    MathSciNet  Article  Google Scholar 

  21. Reiner, V., Signed permutation statistics and cycle type, Europ. J. Combin. 14 (1993), 569–579.

    MathSciNet  Article  Google Scholar 

  22. Robbins, H. “A Remark on Stirling’s Formula.” The American Mathematical Monthly 62, no. 1 (1955), 26–29.

    MathSciNet  MATH  Google Scholar 

  23. Schocker, M., The peak algebra of the symmetric group revisited, Adv. Math. 192 (2005), 259–309.

    MathSciNet  Article  Google Scholar 

  24. Stembridge, J., Enriched \(P\)-partitions, Trans. Amer. Math. Soc. 349 (1997), 763–788.

    MathSciNet  Article  Google Scholar 

  25. Tanny, S., A probabilistic interpretation of Eulerian numbers, Duke Math. J. 40 (1973), 717–722.

    MathSciNet  Article  Google Scholar 

  26. Vershynin, R., High-Dimensional Probability. Cambridge University Press, 2018.

  27. Warren, D. and Seneta, E., Peaks and Eulerian numbers in a random sequence, J. Appl. Probab. 33 (1996), 101–114.

    MathSciNet  Article  Google Scholar 

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We are very grateful to two referees for many detailed comments. Fulman was supported by Simons Foundation Grant 400528.

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Correspondence to Jason Fulman.

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Fulman, J., Kim, G.B. & Lee, S. Central Limit Theorem for Peaks of a Random Permutation in a Fixed Conjugacy Class of \(S_n\). Ann. Comb. 26, 97–123 (2022).

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