A161696 Number of reduced words of length n in the Weyl group B_3.

1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,2

Comments

If the zeros are ignored, this is the coordination sequence for the truncated cuboctahedron (see the Karzes link). - N. J. A. Sloane, Jan 08 2020

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

Table of n, a(n) for n=0..104.

Tom Karzes, Polyhedron Coordination Sequences

Formula

G.f. for B_m is the polynomial Prod_{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^(2k))/(1-x), k=1..3), x, n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 25 2018

Mathematica

CoefficientList[Series[Product[(1-x^(2*k)), {k, 1, 3}] /(1-x)^3, {x, 0, 9}], x] (* G. C. Greubel, Oct 25 2018 *)

Prog

(PARI) t='t+O('t^10); Vec(prod(k=1, 3, 1-t^(2*k))/(1-t)^3) \\ G. C. Greubel, Oct 25 2018
(Magma) m:=10; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // 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: A335894 A101496 A218490 * A196084 A008508 A163301

Adjacent sequences: A161693 A161694 A161695 * A161697 A161698 A161699

Keyword

nonn

Author

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

Status

approved