A162378 Number of reduced words of length n in the Weyl group D_31.

1, 31, 495, 5425, 45879, 319145, 1900920, 9965384, 46909324, 201295028, 796809245, 2937251395, 10161553364, 33205476524, 103050077489, 305131440111, 865481871426, 2359754902590, 6203436293890, 15765840836350, 38828731002622

Offset: 0

Author

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

Keywords

• nonn

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

• Index entries for growth series for groups

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.

Programs

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 = 31}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-10), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

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.