A161410 - OEIS

%I #13 Sep 08 2022 08:45:45

%S 1,7,27,77,183,385,740,1325,2242,3623,5633,8474,12391,17676,24670,

%T 33768,45426,60164,78568,101296,129083,162742,203168,251346,308355,

%U 375369,453663,544620,649732,770602,908952,1066628,1245600,1447967,1675965,1931969,2218494

%N Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

%D N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)

%H Vincenzo Librandi, <a href="/A161410/b161410.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (x^22 + 3*x^21 + 5*x^20 + 7*x^19 + 10*x^18 + 14*x^17 + 17*x^16 + 19*x^15 + 22*x^14 + 25*x^13 + 26*x^12 + 26*x^11 + 26*x^10 + 25*x^9 + 22*x^8 + 19*x^7 + 17*x^6 + 14*x^5 + 10*x^4 + 7*x^3 + 5*x^2 + 3*x + 1)/(x^22 - 4*x^21 + 6*x^20 - 4*x^19 + x^18 - x^15 + 4*x^14 - 6*x^13 + 4*x^12 - 2*x^11 + 4*x^10 - 6*x^9 + 4*x^8 - x^7 + x^4 - 4*x^3 + 6*x^2 - 4*x + 1)

%e Coxeter matrix:

%e . [1 2 3 2 2 2 2]

%e . [2 1 2 3 2 2 3]

%e . [3 2 1 3 2 2 2]

%e . [2 3 3 1 3 2 2]

%e . [2 2 2 3 1 3 2]

%e . [2 2 2 2 3 1 2]

%e . [2 3 2 2 2 2 1]

%t CoefficientList[Series[(x^22 + 3 x^21 + 5 x^20 + 7 x^19 + 10 x^18 + 14 x^17 + 17 x^16 + 19 x^15 + 22 x^14 + 25 x^13 + 26 x^12 + 26 x^11 + 26 x^10 + 25 x^9 + 22 x^8 + 19 x^7 + 17 x^6 + 14 x^5 + 10 x^4 + 7 x^3 + 5 x^2 + 3 x + 1) / (x^22 - 4 x^21 + 6 x^20 - 4 x^19 + x^18 - x^15 + 4 x^14 - 6 x^13 + 4 x^12 - 2 x^11 + 4 x^10 - 6 x^9 + 4 x^8 - x^7 + x^4 - 4 x^3 + 6 x^2 - 4 x + 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)

%o (Magma)

%o Z := Integers();

%o C := SymmetricMatrix(

%o [1,

%o 2,1,

%o 3,2,1,

%o 2,3,3,1,

%o 2,2,2,3,1,

%o 2,2,2,2,3,1,

%o 2,3,2,2,2,2,1]);

%o G := CoxeterGroup(GrpFPCox, C);

%o f := GrowthFunction(G);

%o T<z> := PowerSeriesRing(Z, 50);

%o Eltseq(T!f);

%o // Corrected by _Klaus Brockhaus_, Feb 12 2010

%Y Cf. A161409, A154638.

%K nonn,easy

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Nov 29 2009