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

0,2

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

Table of n, a(n) for n=0..36.

Index entries for sequences related to groups

Formula

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

Example

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]

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 *)

Prog

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

Crossrefs

Cf. A161410, A154638.

Sequence in context: A162209 A161699 A216175 * A002415 A384743 A052515

Adjacent sequences: A161406 A161407 A161408 * A161410 A161411 A161412

Keyword

nonn,fini,full

Author

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

Status

approved