Advertisement

Annals of Combinatorics

, Volume 16, Issue 3, pp 625–650 | Cite as

Generating Functions for Alternating Descents and Alternating Major Index

  • Jeffrey B. RemmelEmail author
Article

Abstract

In 2008, Chebikin introduced the alternating descent set, AltDes(σ), of a permutation σσ 1 ··· σ n in the symmetric group S n as the set of all i such that either i is odd and σ i σ i+1 or i is even and σ i σ i+1. We can then define altdes(σ) = |AltDes(σ)| and \({{\rm altmaj}(\sigma) = \sum_{i \in AltDes(\sigma)}i}\). In this paper, we compute a generating function for the joint distribution of altdes(σ) and altmaj(σ) over S n . Our formula is similar to the formula for the joint distribution of des and maj over the symmetric group that was first proved by Gessel. We also compute similar generating functions for the groups B n and D n and for r-tuples of permutations in S n . Finally we prove a general extension of these formulas in cases where we keep track of descents only at positions r, 2r, . . ..

Mathematics Subject Classification

05A05 05A15 05E05 

Keywords

alternating descents alternating major index symmetric functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adin R.M., Roichman Y.: The flag major index and group actions on polynomial rings. European J. Combin. 22(4), 431–446 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    André, D.: D’eveloppements de secx et tanx. C. R. LXXXVIII, 965–979 (1879)Google Scholar
  3. 3.
    André D.: Mémoire sur les permutations alternées. J. Math. 7, 167–184 (1881)Google Scholar
  4. 4.
    Beck, D.: Permutation enumeration of the symmetric and hyperoctahedral group and the combinatorics of symmetric functions. Ph.D. Thesis, University of California, San Diego (1993)Google Scholar
  5. 5.
    Beck D., Remmel J.: Permutation enumeration of the symmetric group and the combinatorics of symmetric functions. J. Combin. Theory Ser. A 72(1), 1–49 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Beck D.: The combinatorics of symmetric functions and permutation enumeration of the hyperoctahedral group. Discrete Math. 163(1-3), 13–45 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brenti F.: Permutation enumeration, symmetric functions, and unimodality. Pacific J. Math. 157(1), 1–28 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brenti F.: Unimodal polynomials arising from symmetric functions. Proc. Amer. Math. Soc. 108(4), 1133–1141 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Carlitz L.: Sequences and inversions. Duke Math. J. 37, 193–198 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Carlitz L., Scoville R.: Enumeration of pairs of sequences by rises, falls, rising maxima and falling maxima. Acta Math. Hungar. 25, 269–277 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carlitz L., Scoville R., Vaughan T.: Enumeration of pairs of sequences by rises, falls and levels. Manuscripta Math. 19(3), 211–243 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chebikin D.: Variations on descents and inversions in permutations. Electron. J. Combin. 15, #R132 (2008)MathSciNetGoogle Scholar
  13. 13.
    Eğecioğlu Ö., Remmel J.: Brick tabloids and the connection matrices between bases of symmetric functions. Discrete Appl. Math. 34(1-3), 107–120 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fédou J.-M., Rawlings D.: More statistics on permutation pairs. Electron. J. Combin. 1, #R11 (1994)Google Scholar
  15. 15.
    Fédou J.-M., Rawlings D.: Statistics on pairs of permutations. DiscreteMath. 143, 31–45 (1995)zbMATHCrossRefGoogle Scholar
  16. 16.
    Garsia A.M., Gessel I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gessel, I.: Generating functions and enumeration of sequences. Ph.D. Thesis, MIT (1977)Google Scholar
  18. 18.
    MacMahon, P.: Combinatory Analysis, Vols. 1 and 2. Cambridge Univ. Press, Cambridge, (1915); reprinted by Chelsea, New York (1955)Google Scholar
  19. 19.
    Mendes, A.: Building generating functions brick by brick. Ph.D. Thesis, University of California, San Diego (2004)Google Scholar
  20. 20.
    Mendes A., Remmel J.: Descents, inversions, and major indices in permutation groups. Discrete Math. 308(12), 2509–2524 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mendes A., Remmel J., Riehl A.: Permutations with k-regular descent patterns. In: Linton, S., Rus̆kuc, N., Vatter, V. (eds) Permutation Patterns., pp. 259–286. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  22. 22.
    Reiner V.: Signed permutation statistics. European J. Combin. 14(6), 553–567 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La JollaUSA

Personalised recommendations