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Distribution of Descents in Matchings

  • Gene B. KimEmail author
Article
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Abstract

The distribution of descents in certain conjugacy classes of \(S_n\) has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

Keywords

Enumerative combinatorics Probabilistic combinatorics Central limit theorem 

Mathematics Subject Classification

Primary 05A19 60F05 Secondary 60C05 62E20 

Notes

Acknowledgements

The author would like to thank Brendon Rhoades for introducing the author to the world of Young tableaux and helping come up with the bijection in Sect. 3. The author would also like to thank Sangchul Lee for the suggestion of considering moment generating functions instead of characteristic functions and helping with the calculations of bounding the integrals in Sect. 4. Finally, the author would like to thank Jason Fulman for the suggestion of the problem and his guidance.

References

  1. 1.
    Bayer, D., Diaconis, P.: Trailing the dovetail to its lair. Ann. Appl. Probab. 2(2), 294–313 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chatterjee, S., Diaconis, P.: A central limit theorem for a new statistic on permutations. Indian J. Pure Appl. Math. 48(4), 561–573 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, W.Y.C., Deng, E.Y.P., Du, R.R.X., Stanley, R.P., Yan, C.H.: Crossings and nestings of matchings and partitions. Trans. Amer. Math. Soc. 359(4), 1555–1575 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Curtiss, J.H.: A note on the theory of moment generating functions. Ann. Math. Statist. 13, 430–433 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diaconis, P., McGrath, M., Pitman, J.: Riffle shuffles, cycles, and descents. Combinatorica 15(1), 11–29 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diaconis, P., Pitman, J.: Unpublished notes on descentsGoogle Scholar
  7. 7.
    Fulman, J.: The distribution of descents in fixed conjugacy classes of the symmetric group. J. Combin. Theory Ser. A 84(2), 171–180 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fulman, J.: Stein’s method and non-reversible Markov chains. In: Diaconis, P., Holmes, S. (eds.) Stein’s Method: Expository Lectures and Applications, IMS Lecture Notes Monogr. Ser., Vol. 46, pp. 69–77. Inst. Math. Statist., Beachwood, OH (2004)Google Scholar
  9. 9.
    Fulton, W.: Young Tableaux. With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge (1997)Google Scholar
  10. 10.
    Garsia, A.M., Gessel, I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gessel, I.M., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Combin. Theory Ser. A 64(2), 189–215 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldstein, L., Rinott, Y.: A permutation test for matching and its asymptotic distribution. Metron 61(3), 375–388 (2004)MathSciNetGoogle Scholar
  13. 13.
    Harper, L.H.: Stirling behavior is asymptotically normal. Ann. Math. Statist. 38, 410–414 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Knuth, D.E.: The Art of Computer Programming. Vol. 3. Sorting and Searching. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1973)Google Scholar
  15. 15.
    Petersen, T.K.: Eulerian Numbers. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser/Springer, New York (2015)Google Scholar
  16. 16.
    Schützenberger, M.: La correspondance de Robinson. Lecture Notes in Math. 579, 59–113 (1979)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sundaram, S.: The Cauchy identity for \({\rm Sp}(2n)\). J. Combin. Theory Ser. A 53(2), 209–238 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vatutin, V.A.: The numbers of ascending segments in a random permutation and in one inverse to it are asymptotically independent. Discrete Math. Appl. 6(1), 41–52 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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