A161755 Number of reduced words of length n in the Weyl group B_10.

1, 10, 54, 210, 659, 1772, 4235, 9218, 18590, 35178, 63064, 107910, 177297, 281060, 431598, 644136, 936915, 1331286, 1851685, 2525468, 3382588, 4455100, 5776486, 7380800, 9301642, 11570980, 14217849, 17266966, 20737309, 24640716, 28980565

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..100

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..10), x, 101), x, n), n = 0 .. 100); # 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^14) (1 - x^16) (1 - x^18) (1 - x^20) / (1 - x)^10, {x, 0, 100}], x] (* Vincenzo Librandi, Aug 22 2016 *)

Prog

(PARI) t='t+O('t^40); Vec(prod(k=1, 10, 1-t^(2*k))/(1-t)^10) \\ G. C. Greubel, Oct 25 2018
(Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..10]])/(1-t)^10)); // 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: A152762 A161458 A162248 * A053347 A267172 A266764

Adjacent sequences: A161752 A161753 A161754 * A161756 A161757 A161758

Keyword

nonn,easy,fini,full

Author

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

Status

approved