A161409 Number of reduced words of length n in the Weyl group E_6 on 6 generators and order 51840.

1, 6, 20, 50, 105, 195, 329, 514, 754, 1048, 1389, 1765, 2159, 2549, 2911, 3222, 3461, 3611, 3662, 3611, 3461, 3222, 2911, 2549, 2159, 1765, 1389, 1048, 754, 514, 329, 195, 105, 50, 20, 6, 1

Offset: 0

Author

John Cannon and N. J. A. Sloane, Nov 29 2009

Keywords

• nonn
• fini
• full

Examples

			Coxeter matrix:
. [1 2 3 2 2 2]
. [2 1 2 3 2 2]
. [3 2 1 3 2 2]
. [2 3 3 1 3 2]
. [2 2 2 3 1 3]
. [2 2 2 2 3 1]
		

References

• N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)
• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

• Index entries for sequences related to groups

Crossrefs

Cf. A161410, A154638.

Programs

Magma

G := CoxeterGroup(GrpFPCox, "E6");
f := GrowthFunction(G);
Coefficients(PolynomialRing(IntegerRing())!f);
// Corrected by Klaus Brockhaus, Feb 12 2010

Mathematica

CoefficientList[Series[((1-x^2) (1-x^5) (1-x^6) (1-x^8) (1-x^9) (1-x^12))/(1-x)^6,{x,0,40}],x] (* Harvey P. Dale, Aug 17 2011 *)

Formula

G.f.: f(2)f(5)f(6)f(8)f(9)f(12)/f(1)^6 where f(k) = 1-x^k.