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A161696 Number of reduced words of length n in the Weyl group B_3. 22
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Nov 30 2009
STATUS
approved

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Last modified May 23 10:34 EDT 2024. Contains 372760 sequences. (Running on oeis4.)