A162360 Number of reduced words of length n in the Weyl group D_21.

1, 21, 230, 1750, 10394, 51338, 219373, 833129, 2867249, 9073845, 26706681, 73769301, 192639951, 478480891, 1136148412, 2590157556, 5690334091, 12085148911, 24881271591, 49780020675, 96990931345, 184385864444

Offset

0,2

References

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

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Table of n, a(n) for n=0..21.

Index entries for growth series for groups

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

Maple

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k, M) local a, i; global f;
a:=f(k)*mul(f(2*i), i=1..k-1);
seriestolist(series(a, x, M+1));
end proc;

Mathematica

f[m_] := (1-x^m)/(1-x);
With[{k = 21}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k+1), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Crossrefs

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.

Sequence in context: A221037 A221500 A161518 * A161899 A220721 A320588

Adjacent sequences: A162357 A162358 A162359 * A162361 A162362 A162363

Keyword

nonn

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Status

approved