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Number of reduced words of length n in the icosahedral reflection group [3,5] of order 120.
1

%I #18 Feb 16 2025 08:33:11

%S 1,3,5,7,9,11,12,12,12,12,11,9,7,5,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of reduced words of length n in the icosahedral reflection group [3,5] of order 120.

%C This group is also the Weyl group H_3.

%C If the 0's are omitted, this is the coordination sequence for the truncated icosidodecahedron (see Karzes link).

%C Sometimes "great rhombicosidodecahedron" is preferred when referring in particular to the Archimedean polyhedron with this coordination sequence. - _Peter Munn_, Mar 22 2021

%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

%D David Wells, Archimedean polyhedra in Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, 1991, pp. 6-7.

%H Tom Karzes, <a href="/A298808/a298808.html">Polyhedron Coordination Sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreatRhombicosidodecahedron.html">Great Rhombicosidodecahedron</a>

%H <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>

%F G.f.: (1-x^2)*(1-x^6)*(1-x^10)/(1-x)^3.

%o (Magma) G := CoxeterGroup(GrpFPCox, "H3");

%o f := GrowthFunction(G);

%o Coefficients(f);

%Y Cf. A161409, A162493, A162494, A162496, A162497.

%K nonn,fini,changed

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 01 2009