A161755 - OEIS

%I #31 Sep 08 2022 08:45:45

%S 1,10,54,210,659,1772,4235,9218,18590,35178,63064,107910,177297,

%T 281060,431598,644136,936915,1331286,1851685,2525468,3382588,4455100,

%U 5776486,7380800,9301642,11570980,14217849,17266966,20737309,24640716,28980565

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

%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="/A161755/b161755.txt">Table of n, a(n) for n = 0..100</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..10),x,101), x, n), n = 0 .. 100); # _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^14) (1 - x^16) (1 - x^18) (1 - x^20) / (1 - x)^10, {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 22 2016 *)

%o (PARI) t='t+O('t^40); Vec(prod(k=1,10,1-t^(2*k))/(1-t)^10) \\ _G. C. Greubel_, Oct 25 2018

%o (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

%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,easy,fini,full

%O 0,2

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