A162493 -id:A162493 - cp's OEIS frontend

A162494 Number of reduced words of length n in the Weyl group E_8 on 8 generators and order 696729600.

1, 8, 35, 112, 294, 672, 1386, 2640, 4718, 8000, 12978, 20272, 30645, 45016, 64470, 90264, 123829, 166768, 220849, 287992, 370250, 469784, 588833, 729680, 894613, 1085880, 1305640, 1555912, 1838523, 2155056, 2506798, 2894688, 3319268, 3780640, 4278429

Offset: 0

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

N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche VII.)
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Vincenzo Librandi, Table of n, a(n) for n = 0..120

Cf. A161409, A162493.

G := CoxeterGroup(GrpFPCox, "E8");
f := GrowthFunction(G);
Coefficients(f);

CoefficientList[Series[(1 - x^2) (1 - x^8) (1 - x^12) (1 - x^14) (1 - x^18) (1 - x^20) (1 - x^24) (1 - x^30) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

Vec((1-x^2)*(1-x^8)*(1-x^12)*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^24)*(1-x^30)/(1-x)^8 + O(x^121)) \\ Jinyuan Wang, Mar 08 2020

G.f.: (1-x^2)*(1-x^8)*(1-x^12)*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^24)*(1-x^30)/(1-x)^8.

A162496 Number of reduced words of length n in the reflection group [3,4,3] of order 1152.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 48, 60, 71, 80, 87, 92, 94, 92, 87, 80, 71, 60, 48, 36, 25, 16, 9, 4, 1, 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, 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

Offset: 0

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

This is also the Weyl group F_4.

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Cf. A161409, A162493-A162497.

G := CoxeterGroup(GrpFPCox, "F4");
f := GrowthFunction(G);
Coefficients(f);

G.f.: (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/(1-x)^4

A162497 Number of reduced words of length n in the reflection group [3,3,5] of order 14400.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 168, 192, 216, 240, 264, 288, 312, 336, 359, 380, 399, 416, 431, 444, 455, 464, 471, 476, 478, 476, 471, 464, 455, 444, 431, 416, 399, 380, 359, 336, 312, 288, 264, 240, 216, 192, 168, 144, 121, 100, 81, 64, 49, 36, 25, 16

Offset: 0

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

This is also the Weyl group H_4.

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Cf. A161409, A162493-A162496.

G := CoxeterGroup(GrpFPCox, "H4");
f := GrowthFunction(G);
Coefficients(f);

G.f.: (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/(1-x)^4.

A162495 Number of reduced words of length n in the icosahedral reflection group [3,5] of order 120.

Original entry on oeis.org

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, 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, 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

Offset: 0

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

This group is also the Weyl group H_3.
If the 0's are omitted, this is the coordination sequence for the truncated icosidodecahedron (see Karzes link).
Sometimes "great rhombicosidodecahedron" is preferred when referring in particular to the Archimedean polyhedron with this coordination sequence. - Peter Munn, Mar 22 2021

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
David Wells, Archimedean polyhedra in Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, 1991, pp. 6-7.

Tom Karzes, Polyhedron Coordination Sequences
Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron
Index entries for coordination sequences

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

G := CoxeterGroup(GrpFPCox, "H3");
f := GrowthFunction(G);
Coefficients(f);

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

A266785 The growth series for the affine Weyl group E_7.

Original entry on oeis.org

1, 8, 35, 112, 294, 673, 1393, 2668, 4803, 8218, 13476, 21315, 32684, 48782, 71101, 101473, 142121, 195714, 265426, 354999, 468809, 611936, 790238, 1010430, 1280166, 1608124, 2004094, 2479071, 3045353, 3716642, 4508148, 5436696, 6520838, 7780968, 9239441, 10920695, 12851378, 15060479, 17579463, 20442410, 23686158, 27350450, 31478083

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial; also Table 3.1 page 59.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1, 0, 0, 0, 0, 0, 0, 1, -4, 7, -8, 7, -4, 2, -4, 6, -4, 1, 0, 0, -1, 4, -6, 4, -2, 4, -7, 8, -7, 4, -1, 0, 0, 0, 0, 0, 0, 1, -4, 6, -4, 1).

For the growth series for the finite group see A162493.

R:=PowerSeriesRing(Integers(), 45); Coefficients(R!( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*(&*[(1-x^(2*j+6))/(1-x^(2*j+5)): j in [0..4]]) )); // G. C. Greubel, Feb 04 2020

m:=45; S:=series(((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*mul((1-x^(2*j+6))/(1-x^(2*j+5)), j=0..4)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 04 2020

CoefficientList[Series[((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*Product[(1-x^(2*j + 6))/(1-x^(2*j+5)), {j,0,4}], {x,0,45}], x] (* G. C. Greubel, Feb 04 2020 *)

Vec( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*prod(j=0,4, (1-x^(2*j+6))/(1-x^(2*j+5))) +O('x^45) ) \\ G. C. Greubel, Feb 04 2020

def A266785_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*product((1-x^(2*j+6))/(1-x^(2*j+5)) for j in (0..4)) ).list()
A266785_list(45) # G. C. Greubel, Feb 04 2020

G.f.: (1 +t)*(1 +t^3)*(1 +t^5)*(1 +t^7)*(1 +t +t^2 +t^3 +t^4 +t^5 +t^6 +t^7)*(1 +t +t^2 +t^9 +t^10 +t^11)*(1 +t +t^2 +t^3 +t^4 +t^5 +t^6 +t^7 +t^8 +t^9 +t^10 +t^11)/((1-t)^4*(1-t^11)*(1-t^13)*(1-t^17)).

G.f.: ((1-x^2)*(1-x^18)/((1-x)^8*(1-x^17)))*Product_{j=0..4} (1-x^(2*j+6))/(1-x^(2*j+5)). - G. C. Greubel, Feb 05 2020

cp's OEIS Frontend

A162494 Number of reduced words of length n in the Weyl group E_8 on 8 generators and order 696729600.

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A162496 Number of reduced words of length n in the reflection group [3,4,3] of order 1152.

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A162497 Number of reduced words of length n in the reflection group [3,3,5] of order 14400.

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A162495 Number of reduced words of length n in the icosahedral reflection group [3,5] of order 120.

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A266785 The growth series for the affine Weyl group E_7.

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