A162494 Number of reduced words of length n in the Weyl group E_8 on 8 generators and order 696729600.

1, 8, 35, 112, 294, 672, 1386, 2640, 4718, 8000, 12978, 20272, 30645, 45016, 64470, 90264, 123829, 166768, 220849, 287992, 370250, 469784, 588833, 729680, 894613, 1085880, 1305640, 1555912, 1838523, 2155056, 2506798, 2894688, 3319268, 3780640, 4278429

Offset: 0

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

• nonn
• fini
• full

References

• N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche VII.)
• H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..120

Crossrefs

Cf. A161409, A162493.

Programs

Magma

G := CoxeterGroup(GrpFPCox, "E8");
f := GrowthFunction(G);
Coefficients(f);

Mathematica

CoefficientList[Series[(1 - x^2) (1 - x^8) (1 - x^12) (1 - x^14) (1 - x^18) (1 - x^20) (1 - x^24) (1 - x^30) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

PARI

Vec((1-x^2)*(1-x^8)*(1-x^12)*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^24)*(1-x^30)/(1-x)^8 + O(x^121)) \\ Jinyuan Wang, Mar 08 2020

Formula

G.f.: (1-x^2)*(1-x^8)*(1-x^12)*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^24)*(1-x^30)/(1-x)^8.