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A161858 Number of reduced words of length n in the Weyl group B_12.

%I A161858 #26 Mar 21 2025 15:55:37
%S A161858 1,12,77,352,1286,3992,10933,27092,61841,131768,264759,505660,923858,
%T A161858 1623116,2753972,4528964,7240871,11284064,17178942,25599288,37402222,
%U A161858 53660256,75694775,105110084,143826980,194114636,258619428,340389204,442891395,570023312,726112969
%N A161858 Number of reduced words of length n in the Weyl group B_12.
%C A161858 Computed with MAGMA using commands similar to those used to compute A161409.
%D A161858 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%D A161858 N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)
%H A161858 G. C. Greubel, <a href="/A161858/b161858.txt">Table of n, a(n) for n = 0..144</a>
%F A161858 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 A161858 seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..12),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 25 2018
%t A161858 CoefficientList[Series[Product[(1-x^(2*k)),{k,1,12}]/(1-x)^12,{x,0,50}], x] (* _G. C. Greubel_, Oct 25 2018 *)
%o A161858 (PARI) t='t+O('t^50); Vec(prod(k=1,12,1-t^(2*k))/(1-t)^12) \\ _G. C. Greubel_, Oct 25 2018
%o A161858 (Magma) m:=50; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..12]])/(1-t)^12)); // _G. C. Greubel_, Oct 25 2018
%Y A161858 Row n=12 of A128084.
%Y A161858 The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.
%K A161858 nonn,fini,full
%O A161858 0,2
%A A161858 _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009