A162212 Number of reduced words of length n in the Weyl group D_9.

1, 9, 44, 156, 449, 1113, 2463, 4983, 9372, 16587, 27877, 44802, 69231, 103314, 149425, 210075, 287796, 384999, 503812, 645906, 812319, 1003290, 1218116, 1455045, 1711216, 1982655, 2264333, 2550288, 2833809, 3107676, 3364445

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..30.

Index entries for growth series for groups

Formula

The growth series for D_k is the polynomial f(k)*Prod_{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

A162212g := proc(m::integer)
    (1-x^m)/(1-x) ;
end proc:
A162212 := proc(n, k)
    g := A162212g(k);
    for m from 2 to 2*k-2 by 2 do
        g := g*A162212g(m) ;
    end do:
    g := expand(g) ;
    coeftayl(g, x=0, n) ;
end proc:
seq( A162212(n, 9), n=0..30) ; # R. J. Mathar, Jan 19 2016
# 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

n = 9;
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

The growth series for D_k, k >= 3, are also the rows of the triangle A162206.

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: A034194 A075206 A161457 * A161733 A374931 A050486

Adjacent sequences: A162209 A162210 A162211 * A162213 A162214 A162215

Keyword

nonn

Author

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

Status

approved