A161716 Number of reduced words of length n in the Weyl group B_7.

1, 7, 27, 77, 181, 371, 686, 1170, 1869, 2827, 4082, 5662, 7581, 9835, 12399, 15225, 18242, 21358, 24464, 27440, 30162, 32510, 34376, 35672, 36336, 36336, 35672, 34376, 32510, 30162, 27440, 24464, 21358, 18242, 15225, 12399, 9835, 7581, 5662

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..49

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:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..7]])/(1-t)^7)); // G. C. Greubel, Oct 25 2018

Maple

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..7),x,n+1), x, n), n = 0 .. 40); # 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)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2016 *)

PARI

t='t+O('t^50); Vec(prod(k=1,7,1-t^(2*k))/(1-t)^7) \\ 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.