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, 3596763, 3797695, 3961044, 4081645

Offset: 0

Author

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

Keywords

• nonn
• fini
• full

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

• Andrew Howroyd, Table of n, a(n) for n = 0..72
• Index entries for growth series for groups

Crossrefs

Row 9 of 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.

Programs

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 *)

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.