cp's OEIS Frontend

A161409 Number of reduced words of length n in the Weyl group E_6 on 6 generators and order 51840.

%I A161409 #18 Sep 08 2022 08:45:45
%S A161409 1,6,20,50,105,195,329,514,754,1048,1389,1765,2159,2549,2911,3222,
%T A161409 3461,3611,3662,3611,3461,3222,2911,2549,2159,1765,1389,1048,754,514,
%U A161409 329,195,105,50,20,6,1
%N A161409 Number of reduced words of length n in the Weyl group E_6 on 6 generators and order 51840.
%D A161409 N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)
%D A161409 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
%H A161409 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F A161409 G.f.: f(2)f(5)f(6)f(8)f(9)f(12)/f(1)^6 where f(k) = 1-x^k.
%e A161409 Coxeter matrix:
%e A161409 . [1 2 3 2 2 2]
%e A161409 . [2 1 2 3 2 2]
%e A161409 . [3 2 1 3 2 2]
%e A161409 . [2 3 3 1 3 2]
%e A161409 . [2 2 2 3 1 3]
%e A161409 . [2 2 2 2 3 1]
%t A161409 CoefficientList[Series[((1-x^2) (1-x^5) (1-x^6) (1-x^8) (1-x^9) (1-x^12))/(1-x)^6,{x,0,40}],x] (* _Harvey P. Dale_, Aug 17 2011 *)
%o A161409 (Magma)
%o A161409 G := CoxeterGroup(GrpFPCox, "E6");
%o A161409 f := GrowthFunction(G);
%o A161409 Coefficients(PolynomialRing(IntegerRing())!f);
%o A161409 // Corrected by _Klaus Brockhaus_, Feb 12 2010
%Y A161409 Cf. A161410, A154638.
%K A161409 nonn,fini,full
%O A161409 0,2
%A A161409 _John Cannon_ and _N. J. A. Sloane_, Nov 29 2009