A161410 Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

1, 7, 27, 77, 183, 385, 740, 1325, 2242, 3623, 5633, 8474, 12391, 17676, 24670, 33768, 45426, 60164, 78568, 101296, 129083, 162742, 203168, 251346, 308355, 375369, 453663, 544620, 649732, 770602, 908952, 1066628, 1245600, 1447967, 1675965, 1931969, 2218494

Offset: 0

Author

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

Keywords

• nonn
• easy

Examples

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

References

• N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A161409, A154638.

Programs

Magma

Z := Integers();
C := SymmetricMatrix(
[1,
2,1,
3,2,1,
2,3,3,1,
2,2,2,3,1,
2,2,2,2,3,1,
2,3,2,2,2,2,1]);
G := CoxeterGroup(GrpFPCox, C);
f := GrowthFunction(G);
T := PowerSeriesRing(Z, 50);
Eltseq(T!f);
// Corrected by Klaus Brockhaus, Feb 12 2010

Mathematica

CoefficientList[Series[(x^22 + 3 x^21 + 5 x^20 + 7 x^19 + 10 x^18 + 14 x^17 + 17 x^16 + 19 x^15 + 22 x^14 + 25 x^13 + 26 x^12 + 26 x^11 + 26 x^10 + 25 x^9 + 22 x^8 + 19 x^7 + 17 x^6 + 14 x^5 + 10 x^4 + 7 x^3 + 5 x^2 + 3 x + 1) / (x^22 - 4 x^21 + 6 x^20 - 4 x^19 + x^18 - x^15 + 4 x^14 - 6 x^13 + 4 x^12 - 2 x^11 + 4 x^10 - 6 x^9 + 4 x^8 - x^7 + x^4 - 4 x^3 + 6 x^2 - 4 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

Formula

G.f.: (x^22 + 3*x^21 + 5*x^20 + 7*x^19 + 10*x^18 + 14*x^17 + 17*x^16 + 19*x^15 + 22*x^14 + 25*x^13 + 26*x^12 + 26*x^11 + 26*x^10 + 25*x^9 + 22*x^8 + 19*x^7 + 17*x^6 + 14*x^5 + 10*x^4 + 7*x^3 + 5*x^2 + 3*x + 1)/(x^22 - 4*x^21 + 6*x^20 - 4*x^19 + x^18 - x^15 + 4*x^14 - 6*x^13 + 4*x^12 - 2*x^11 + 4*x^10 - 6*x^9 + 4*x^8 - x^7 + x^4 - 4*x^3 + 6*x^2 - 4*x + 1)