A162403 Number of reduced words of length n in the Weyl group D_41.

1, 41, 860, 12300, 134889, 1209377, 9230207, 61657399, 367846424, 1990342376, 9885562358, 45508669878, 195729780567, 791712506207, 3028721321382, 11010682764150, 38197208930405, 126905454993645, 405078061871575

Offset

0,2

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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

Robert Israel, Table of n, a(n) for n = 0..1640

Index entries for growth series for groups

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

Maple

f:= k -> 1-x^k:
g:= n -> f(n)*mul(f(2*i), i=1..n-1)/f(1)^n:
S:= expand(normal(g(41))):
seq(coeff(S, x, j), j=0..degree(S, x)); # Robert Israel, Oct 07 2015

Mathematica

n = 41;
x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

Crossrefs

Cf. A161409.

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: A299332 A161662 A162178 * A010993 A208431 A275355

Adjacent sequences: A162400 A162401 A162402 * A162404 A162405 A162406

Keyword

nonn,fini,full

Author

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

Status

approved