A162366 Number of reduced words of length n in the Weyl group D_24.
1, 24, 299, 2576, 17249, 95656, 457170, 1934920, 7396155, 25914720, 84197296, 256013184, 734002335, 1996645640, 5180091511, 12874497504, 30770197710, 70952341040, 158302199085, 342599792520, 720836052690, 1477396844040, 2954878145505, 5776377855120, 11052719207368
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 24 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
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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;
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Mathematica
f[m_] := (1-x^m)/(1-x); With[{k = 24}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-2), 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