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

Author

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

Keywords

• nonn

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

• Tom Karzes, Polyhedron Coordination Sequences

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.

Programs

Magma

m:=10; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // G. C. Greubel, Oct 25 2018

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 *)

PARI

t='t+O('t^10); Vec(prod(k=1,3,1-t^(2*k))/(1-t)^3) \\ G. C. Greubel, Oct 25 2018

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.