A161733 Number of reduced words of length n in the Weyl group B_9.

1, 9, 44, 156, 449, 1113, 2463, 4983, 9372, 16588, 27886, 44846, 69387, 103763, 150538, 212538, 292779, 394371, 520399, 673783, 857121, 1072521, 1321430, 1604470, 1921291, 2270451, 2649332, 3054100, 3479715, 3919995, 4367735

Offset: 0

Author

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

Keywords

• nonn
• easy
• fini
• full

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..81

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:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..9]])/(1-t)^9)); // G. C. Greubel, Oct 25 2018

Maple

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..9),x,n+1), x, n), n = 0 .. 30); # 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)^9, {x, 0, 81}], x] (* Vincenzo Librandi, Aug 22 2016 *)

PARI

t='t+O('t^40); Vec(prod(k=1,9,1-t^(2*k))/(1-t)^9) \\ G. C. Greubel, Oct 25 2018

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.