A161699 Number of reduced words of length n in the Weyl group B_6.

1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919, 2254, 2565, 2832, 3037, 3166, 3210, 3166, 3037, 2832, 2565, 2254, 1919, 1580, 1255, 958, 699, 484, 315, 190, 104, 50, 20, 6, 1

Offset

0,2

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

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

Links

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

Formula

G.f. for B_m is the polynomial Product_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

Maple

seq(coeff(series(mul((1-x^(2*k))/(1-x), k=1..6), x, n+1), x, n), n = 0 .. 36); # Muniru A Asiru, Oct 25 2018

Mathematica

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) / (1 - x)^6, {x, 0, 50}], x]  (* Vincenzo Librandi, Aug 22 2016 *)

Prog

(PARI) t='t+O('t^40); Vec(prod(k=1, 6, 1-t^(2*k))/(1-t)^6) \\ G. C. Greubel, Oct 25 2018
(Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..6]])/(1-t)^6)); // G. C. Greubel, Oct 25 2018

Crossrefs

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

Sequence in context: A063488 A299292 A162209 * A216175 A161409 A002415

Adjacent sequences: A161696 A161697 A161698 * A161700 A161701 A161702

Keyword

nonn,fini,full

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Status

approved