A162364 Number of reduced words of length n in the Weyl group D_22.
1, 22, 252, 2002, 12396, 63734, 283107, 1116236, 3983485, 13057330, 39764011, 113533312, 306173263, 784654154, 1920802566, 4510960122, 10201294213, 22286443124, 47167714715, 96947735390, 193938666735, 378324531180, 720920510114, 1344018408128, 2454841642382
Offset: 0
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
Crossrefs
Row 22 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
# 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 = 22}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^k, 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.
Comments