cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A162493 Number of reduced words of length n in the Weyl group E_7 on 7 generators and order 2903040.

Original entry on oeis.org

1, 7, 27, 77, 182, 378, 713, 1247, 2051, 3205, 4795, 6909, 9632, 13040, 17194, 22134, 27874, 34398, 41657, 49567, 58009, 66831, 75852, 84868, 93659, 101997, 109655, 116417, 122087, 126497, 129514, 131046, 131046, 129514, 126497, 122087, 116417, 109655
Offset: 0

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Author

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

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche VI.)
  • 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.

Links

Crossrefs

Cf. A161409, A162494.

Programs

  • Magma
    G := CoxeterGroup(GrpFPCox, "E7");
f := GrowthFunction(G);
Coefficients(f);
  • PARI
    Vec((1-x^2)*(1-x^6)*(1-x^8)*(1-x^10)*(1-x^12)*(1-x^14)*(1-x^18)/(1-x)^7 + O(x^64)) \\ Jinyuan Wang, Mar 08 2020

Formula

G.f.: (1-x^2)*(1-x^6)*(1-x^8)*(1-x^10)*(1-x^12)*(1-x^14)*(1-x^18)/(1-x)^7.

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

Views

Author

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

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

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

Programs

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

Formula

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

A267176 The growth series for the affine Weyl group E_8.

Original entry on oeis.org

1, 9, 44, 156, 450, 1122, 2508, 5149, 9875, 17910, 31000, 51567, 82892, 129330, 196561, 291880, 424528, 606067, 850803, 1176260, 1603708, 2158748, 2871957, 3779597, 4924393, 6356383, 8133842, 10324283, 13005538, 16266923, 20210492, 24952383, 30624256, 37374826, 45371496, 54802094, 65876718, 78829693, 93921640, 111441659, 131709633
Offset: 0

Views

Author

N. J. A. Sloane, Jan 11 2016

Keywords

References

  • 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.

Links

  • Ray Chandler, Table of n, a(n) for n = 0..1000
  • Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 2, -2, 1, 0, 1, -2, 2, -2, 1, 0, 1, -3, 3, -1, 0, -1, 3, -5, 5, -3, 2, -2, 2, -3, 4, -2, 0, -2, 5, -6, 6, -4, 1, -1, 4, -6, 5, -3, 2, -3, 6, -7, 4, -2, 2, -2, 4, -7, 6, -3, 2, -3, 5, -6, 4, -1, 1, -4, 6, -6, 5, -2, 0, -2, 4, -3, 2, -2, 2, -3, 5, -5, 3, -1, 0, -1, 3, -3, 1, 0, 1, -2, 2, -2, 1, 0, 1, -2, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1).

Crossrefs

For the growth series for the finite group see A162494.

Formula

G.f. = t1/t2, where t1 is
(1+t)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)
*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+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+t^12+t^13)
*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17)
*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17+t^18+t^19)
*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9+t^10+t^11+t^12+t^13+t^14+t^15+t^16+t^17+t^18+t^19+t^20+t^21+t^22+t^23)
*(1+t^20+t^21+t^22+t^23+t^18+t^19+t^2+t^3+t^4+t^5+t^6+t^12+t^13+t^7+t^8+t^9+t^10+t^11+t^14+t^15+t^16+t^17+t+t^24+t^25+t^26+t^27+t^28+t^29),
and t2 = (1-t)*(1-t^7)*(1-t^11)*(1-t^13)*(1-t^17)*(1-t^19)*(1-t^23)*(1-t^29).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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