Annals of Combinatorics

, Volume 16, Issue 3, pp 625–650

# Generating Functions for Alternating Descents and Alternating Major Index

• Jeffrey B. Remmel
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

## 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)
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)
6. 6.
Beck D.: The combinatorics of symmetric functions and permutation enumeration of the hyperoctahedral group. Discrete Math. 163(1-3), 13–45 (1997)
7. 7.
Brenti F.: Permutation enumeration, symmetric functions, and unimodality. Pacific J. Math. 157(1), 1–28 (1993)
8. 8.
Brenti F.: Unimodal polynomials arising from symmetric functions. Proc. Amer. Math. Soc. 108(4), 1133–1141 (1990)
9. 9.
Carlitz L.: Sequences and inversions. Duke Math. J. 37, 193–198 (1970)
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)
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)
12. 12.
Chebikin D.: Variations on descents and inversions in permutations. Electron. J. Combin. 15, #R132 (2008)
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)
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)
16. 16.
Garsia A.M., Gessel I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)
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)
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)
22. 22.
Reiner V.: Signed permutation statistics. European J. Combin. 14(6), 553–567 (1993)