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A161699 Number of reduced words of length n in the Weyl group B_6. 22


%S 1,6,20,50,104,190,315,484,699,958,1255,1580,1919,2254,2565,2832,3037,

%T 3166,3210,3166,3037,2832,2565,2254,1919,1580,1255,958,699,484,315,

%U 190,104,50,20,6,1

%N Number of reduced words of length n in the Weyl group B_6.

%C Computed with MAGMA using commands similar to those used to compute A161409.

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

%D N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

%H G. C. Greubel, <a href="/A161699/b161699.txt">Table of n, a(n) for n = 0..36</a>

%F 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.

%p seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..6),x,n+1), x, n), n = 0 .. 36); # _Muniru A Asiru_, Oct 25 2018

%t CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) / (1 - x)^6, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 22 2016 *)

%o (PARI) t='t+O('t^40); Vec(prod(k=1,6,1-t^(2*k))/(1-t)^6) \\ _G. C. Greubel_, Oct 25 2018

%o (MAGMA) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..6]])/(1-t)^6)); // _G. C. Greubel_, Oct 25 2018

%Y 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.

%K nonn,fini,full

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)