## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### The $q$-Eulerian distribution of the elliptic Weyl group of type $A_1^{(1,1)}$

#### Abstract

We calculate the $q$-Eulerian distribution $W(t,q)$ of the elliptic Weyl group of type $A_1^{(1,1)}$, which is a formal power series in $\mathbf{Z}[[t,q]]$, and classically defined for any Coxeter system $(W,S)$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 3 (2006), 53-55.

Dates
First available in Project Euclid: 4 April 2006

https://projecteuclid.org/euclid.pja/1144158994

Digital Object Identifier
doi:10.3792/pjaa.82.53

Mathematical Reviews number (MathSciNet)
MR2214775

Zentralblatt MATH identifier
1111.20033

Subjects
Primary: 20D30: Series and lattices of subgroups

#### Citation

Takebayashi, Tadayoshi. The $q$-Eulerian distribution of the elliptic Weyl group of type $A_1^{(1,1)}$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 3, 53--55. doi:10.3792/pjaa.82.53. https://projecteuclid.org/euclid.pja/1144158994

#### References

• K. Saito and T. Takebayashi, Extended affine root systems. III. Elliptic Weyl groups, Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 301–329.
• V. Reiner, The distribution of descent and length in a Coxeter group, Electron. J. Combin. 2 (1995), R 25. (Electronic).
• R. P. Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 336–356.
• M. Wakimoto Poincaré series of the Weyl group of elliptic Lie algebras $A_1^{(1,1)}$ and $A_1^{(1,1)*}$. (Preprint). q-alg/9705025.
• T. Takebayashi, Poincaré series of the Weyl groups of the elliptic root systems $A\sb 1\sp {(1,1)},\ A\sb 1\sp {{(1,1)}\sp *}$ and $A\sb 2\sp {(1,1)}$, J. Algebraic Combin. 17 (2003), no. 3, 211–223.