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Number of reduced words of length n in the Weyl group B_5.
0

%I #19 Sep 08 2022 08:45:45

%S 1,5,14,30,54,86,125,169,215,259,297,325,340,340,325,297,259,215,169,

%T 125,86,54,30,14,5,1

%N Number of reduced words of length n in the Weyl group B_5.

%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.)

%F G.f. for B_m is the polynomial Prod_{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..5),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Oct 25 2018

%t CoefficientList[Series[Product[(1-x^(2*k)), {k,1,5}] /(1-x)^5, {x,0,25}], x] (* _G. C. Greubel_, Oct 25 2018 *)

%o (PARI) t='t+O('t^26); Vec(prod(k=1,5,1-t^(2*k))/(1-t)^5) \\ _G. C. Greubel_, Oct 25 2018

%o (Magma) m:=26; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..5]])/(1-t)^5)); // _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