login
Number of reduced words of length n in the Weyl group B_7.
22

%I #28 Sep 08 2022 08:45:45

%S 1,7,27,77,181,371,686,1170,1869,2827,4082,5662,7581,9835,12399,15225,

%T 18242,21358,24464,27440,30162,32510,34376,35672,36336,36336,35672,

%U 34376,32510,30162,27440,24464,21358,18242,15225,12399,9835,7581,5662

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

%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="/A161716/b161716.txt">Table of n, a(n) for n = 0..49</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^(2k))/(1-x),k=1..7),x,n+1), x, n), n = 0 .. 40); # _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)^7, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 22 2016 *)

%o (PARI) t='t+O('t^50); Vec(prod(k=1,7,1-t^(2*k))/(1-t)^7) \\ _G. C. Greubel_, Oct 25 2018

%o (Magma) m:=50; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..7]])/(1-t)^7)); // _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