A162208 Number of reduced words of length n in the Weyl group D_5.

1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1

Offset: 0

Author

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

Keywords

• nonn
• fini
• full

References

• N. Bourbaki, Groupes et algèbres 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

• Index entries for growth series for groups

Crossrefs

Row 5 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

A162208g := proc(m::integer)
    (1-x^m)/(1-x) ;
end proc:
A162208 := proc(n,k)
    g := A162208g(k);
    for m from 2 to 2*k-2 by 2 do
        g := g*A162208g(m) ;
    end do:
    g := expand(g) ;
    coeftayl(g,x=0,n) ;
end proc:
seq( A162208(n,5),n=0..60) ; # R. J. Mathar, Jan 19 2016

Mathematica

n = 5;
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.