A162370 Number of reduced words of length n in the Weyl group D_28.

1, 28, 405, 4032, 31058, 197288, 1075697, 5174180, 22396787, 88562288, 323686749, 1103799420, 3538931886, 10735761372, 30981056614, 85436083852, 226032307036, 575653531156, 1415485760287, 3369314791024, 7781762472652, 17474847498496

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 = 28}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-6), 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: A125463 A161571 A161956 * A162727 A010980 A022592

Adjacent sequences: A162367 A162368 A162369 * A162371 A162372 A162373

Keyword

nonn

Author

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

Status

approved