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.

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A161878 Number of reduced words of length n in the Weyl group B_18.

Original entry on oeis.org

1, 18, 170, 1122, 5813, 25176, 94791, 318630, 974643, 2752112, 7253764, 18003544, 42378246, 95162260, 204856291, 424515042, 849825768, 1648470894, 3106669575, 5701318544, 10209535182, 17871860844, 30631158960, 51476623220, 84931612739
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=18 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161880 Number of reduced words of length n in the Weyl group B_20.

Original entry on oeis.org

1, 20, 209, 1520, 8644, 40944, 168035, 613756, 2034120, 6206596, 17632836, 47062620, 118870650, 285840940, 657667521, 1454009144, 3100176535, 6394814820, 12796122680, 24898749084, 47210910670, 87394933100, 158210114490, 280501919100, 487725336449, 832684355656
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=20 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161899 Number of reduced words of length n in the Weyl group B_21.

Original entry on oeis.org

1, 21, 230, 1750, 10394, 51338, 219373, 833129, 2867249, 9073845, 26706681, 73769301, 192639951, 478480891, 1136148412, 2590157556, 5690334091, 12085148911, 24881271591, 49780020675, 96990931345, 184385864445, 342595978935, 623097898035, 1110823234484, 1943507590140
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=21 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161954 Number of reduced words of length n in the Weyl group B_27.

Original entry on oeis.org

1, 27, 377, 3627, 27026, 166230, 878409, 4098483, 17222607, 66165501, 235124461, 780112671, 2435132466, 7196829486, 20245295242, 54455027238, 140596223184, 349621224120, 839832229131, 1953829030737, 4412447681628, 9693085025844, 20750619208890, 43361428085886
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=27 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161956 Number of reduced words of length n in the Weyl group B_28.

Original entry on oeis.org

1, 28, 405, 4032, 31058, 197288, 1075697, 5174180, 22396787, 88562288, 323686749, 1103799420, 3538931886, 10735761372, 30981056614, 85436083852, 226032307036, 575653531156, 1415485760287, 3369314791024, 7781762472652, 17474847498496, 38225466707386, 81586894793272
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=28 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161972 Number of reduced words of length n in the Weyl group B_29.

Original entry on oeis.org

1, 29, 434, 4466, 35524, 232812, 1308509, 6482689, 28879476, 117441764, 441128513, 1544927933, 5083859819, 15819621191, 46800677805, 132236761657, 358269068693, 933922599849, 2349408360136, 5718723151160, 13500485623812, 30975333122308, 69200799829694, 150787694622966
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=29 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161976 Number of reduced words of length n in the Weyl group B_30.

Original entry on oeis.org

1, 30, 464, 4930, 40454, 273266, 1581775, 8064464, 36943940, 154385704, 595514217, 2140442150, 7224301969, 23043923160, 69844600965, 202081362622, 560350431315, 1494273031164, 3843681391300, 9562404542460, 23062890166272, 54038223288580, 123239023118274, 274026717741240
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Nov 30 2009

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Links

Crossrefs

Row n=30 of A128084.

Formula

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A267168 Growth series for affine Coxeter group B_5.

Original entry on oeis.org

1, 6, 20, 51, 110, 211, 372, 615, 966, 1455, 2117, 2991, 4120, 5551, 7334, 9524, 12180, 15365, 19146, 23594, 28784, 34795, 41711, 49619, 58611, 68783, 80234, 93067, 107389, 123312, 140952, 160430, 181870, 205400, 231152, 259261, 289867, 323114, 359151, 398131, 440211, 485551, 534315, 586672, 642794, 702858, 767045, 835540, 908532, 986214
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).

Links

Crossrefs

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

Formula

The growth series for the affine Coxeter group of type B_k (k >= 2) has g.f. = Product_i (1-x^{m_i+1})/((1-x)*(1-x^{m_i})) where the m_i are [1,3,5,...,2k-1].
Here (k=5) the G.f. is -(1+t)*(1+t+t^2+t^3)*(t^3+1)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^5+1)/(-1+t^9)/(-1+t^7)/(-1+t)^3.

A267169 Growth series for affine Coxeter group B_6.

Original entry on oeis.org

1, 7, 27, 78, 188, 399, 771, 1386, 2352, 3807, 5924, 8916, 13041, 18606, 25971, 35554, 47835, 63361, 82750, 106695, 135968, 171425, 214011, 264764, 324820, 395417, 477900, 573724, 684459, 811795, 957546, 1123655, 1312198, 1525389, 1765583, 2035281, 2337134, 2673948, 3048689, 3464488, 3924646, 4432636, 4992108, 5606893, 6281008, 7018660
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).

Links

Crossrefs

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

Formula

The growth series for the affine Coxeter group of type B_k (k >= 2) has g.f. = Product_i (1-x^{m_i+1})/((1-x)*(1-x^{m_i})) where the m_i are [1,3,5,...,2k-1].

A267170 Growth series for affine Coxeter group B_7.

Original entry on oeis.org

1, 8, 35, 113, 301, 700, 1471, 2857, 5209, 9016, 14940, 23856, 36897, 55504, 81481, 117055, 164941, 228412, 311373, 418440, 555023, 727414, 942880, 1209761, 1537573, 1937115, 2420581, 3001676, 3695738, 4519865, 5493047, 6636302, 7972817, 9528094, 11330100, 13409422, 15799426, 18536422, 21659833, 25212370, 29240211, 33793185, 38924961
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).

Links

Crossrefs

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

Formula

The growth series for the affine Coxeter group of type B_k (k >= 2) has g.f. = Product_i (1-x^{m_i+1})/((1-x)*(1-x^{m_i})) where the m_i are [1,3,5,...,2k-1].
Previous Showing 31-40 of 66 results. Next

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

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