A128084 -id:A128084 - cp's OEIS frontend

A267171 Growth series for affine Coxeter group B_8.

1, 9, 44, 157, 458, 1158, 2629, 5486, 10695, 19711, 34651, 58507, 95404, 150908, 232389, 349445, 514393, 742832, 1054283, 1472911, 2028333, 2756518, 3700784, 4912897, 6454277, 8397316, 10826813, 13841530, 17555875, 22101717, 27630339, 34314534, 42350849, 51961982, 63399337, 76945741, 92918329, 111671603, 133600669, 159144658, 188790335

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).

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 9, 0, -9, 9, 0, -9, 10, -5, 2, -5, 11, -14, 11, -5, 1, 0, 0, -1, 5, -11, 14, -12, 10, -12, 14, -11, 5, -1, 0, 0, 1, -5, 11, -14, 11, -5, 2, -5, 10, -9, 0, 9, -9, 0, 9, -10, 5, -1).

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.

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].

A267172 Growth series for affine Coxeter group B_9.

Original entry on oeis.org

1, 10, 54, 211, 669, 1827, 4456, 9942, 20637, 40348, 74999, 133506, 228910, 379818, 612207, 961652, 1476045, 2218878, 3273169, 4746115, 6774561, 9531380, 13232864, 18147232, 24604366, 33006891, 43842720, 57699190, 75278921, 97417535, 125103378, 159499393, 201967298, 254094228, 317722005, 394979205, 488316197, 600543335, 734872490

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).

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 19, -9, -9, 18, -9, -9, 18, -9, -8, 12, 7, -34, 43, -25, -2, 11, 6, -28, 27, 0, -27, 26, 6, -43, 52, -26, -5, 5, 26, -52, 43, -6, -26, 27, 0, -27, 28, -6, -11, 2, 25, -43, 34, -7, -12, 8, 9, -18, 9, 9, -18, 9, 9, -19, 15, -6, 1).

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.

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].

A267173 Growth series for affine Coxeter group B_10.

Original entry on oeis.org

1, 11, 65, 276, 945, 2772, 7228, 17170, 37807, 78155, 153154, 286660, 515570, 895388, 1507595, 2469247, 3945292, 6164170, 9437339, 14183455, 20958025, 30489449, 43722470, 61870160, 86475684, 119485204, 163333410, 221043295, 296341927, 393794113, 518955998, 678550795, 880669001, 1134995618, 1453067068, 1848560666, 2337619696, 2939217322

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).

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 34, -28, -1, 34, -48, 34, -1, -28, 35, -28, 29, -42, 49, -33, -5, 42, -46, 8, 43, -66, 43, 7, -39, 21, 29, -62, 55, -30, 22, -44, 74, -71, 18, 52, -84, 52, 18, -71, 74, -44, 22, -30, 55, -62, 29, 21, -39, 7, 43, -66, 43, 8, -46, 42, -5, -33, 49, -42, 29, -28, 35, -28, -1, 34, -48, 34, -1, -28, 34, -21, 7, -1).

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.

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].

A267174 Growth series for affine Coxeter group B_11.

Original entry on oeis.org

1, 12, 77, 353, 1298, 4070, 11298, 28468, 66275, 144430, 297584, 584244, 1099814, 1995202, 3502797, 5972044, 9917336, 16081506, 25518845, 39702300, 60660325, 91149775, 134872255, 196742469, 283218364, 402704237, 566039474, 787087225, 1083439094, 1477253844, 1996250190, 2674875984, 3555678491, 4690903019, 6144349905, 7993522778

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).

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 55, -62, 27, 35, -82, 82, -35, -27, 62, -55, 29, -16, 29, -55, 63, -35, -6, 19, 9, -55, 82, -62, 0, 62, -82, 54, 0, -56, 98, -120, 126, -120, 98, -57, 8, 26, -27, -1, 35, -55, 55, -35, 1, 27, -26, -8, 57, -98, 120, -126, 120, -98, 56, 0, -54, 82, -62, 0, 62, -82, 55, -9, -19, 6, 35, -63, 55, -29, 16, -29, 55, -62, 27, 35, -82, 82, -35, -27, 62, -55, 28, -8, 1).

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.

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].

A161697 Number of reduced words of length n in the Weyl group B_4.

Original entry on oeis.org

1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

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

nonn

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

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

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.

m:=17; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..4]])/(1-t)^4)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..4),x,n+1), x, n), n = 0 .. 100); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,4}] /(1-x)^4, {x,0,16}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^17); Vec(prod(k=1,4,1-t^(2*k))/(1-t)^4) \\ G. C. Greubel, Oct 25 2018

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.

A161698 Number of reduced words of length n in the Weyl group B_5.

Original entry on oeis.org

1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1

Offset: 0

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

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

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

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.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..5]])/(1-t)^5)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..5),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,5}] /(1-x)^5, {x,0,25}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^26); Vec(prod(k=1,5,1-t^(2*k))/(1-t)^5) \\ G. C. Greubel, Oct 25 2018

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.

A161977 Number of reduced words of length n in the Weyl group B_31.

Original entry on oeis.org

1, 31, 495, 5425, 45879, 319145, 1900920, 9965384, 46909324, 201295028, 796809245, 2937251395, 10161553364, 33205476524, 103050077489, 305131440111, 865481871426, 2359754902590, 6203436293890, 15765840836350, 38828731002622

Offset: 0

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

nonn

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

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.)

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.

A161987 Number of reduced words of length n in the Weyl group B_32.

Original entry on oeis.org

1, 32, 527, 5952, 51831, 370976, 2271896, 12237280, 59146604, 260441632, 1057250877, 3994502272, 14156055636, 47361532160, 150411609649, 455543049760, 1321024921186, 3680779823776, 9884216117666, 25650056954016

Offset: 0

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

nonn

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

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.)

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.

A161988 Number of reduced words of length n in the Weyl group B_33.

Original entry on oeis.org

1, 33, 560, 6512, 58343, 429319, 2701215, 14938495, 74085099, 334526731, 1391777608, 5386279880, 19542335516, 66903867676, 217315477325, 672858527085, 1993883448271, 5674663272047, 15558879389713, 41208936343729

Offset: 0

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

nonn

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

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.)

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.

A161991 Number of reduced words of length n in the Weyl group B_34.

Original entry on oeis.org

1, 34, 594, 7106, 65449, 494768, 3195983, 18134478, 92219577, 426746308, 1818523916, 7204803796, 26747139312, 93651006988, 310966484313, 983825011398, 2977708459669, 8652371731716, 24211251121429, 65420187465158

Offset: 0

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

nonn

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

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.)

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.

cp's OEIS Frontend

A267171 Growth series for affine Coxeter group B_8.

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A267172 Growth series for affine Coxeter group B_9.

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A267173 Growth series for affine Coxeter group B_10.

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A267174 Growth series for affine Coxeter group B_11.

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A161697 Number of reduced words of length n in the Weyl group B_4.

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A161698 Number of reduced words of length n in the Weyl group B_5.

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A161977 Number of reduced words of length n in the Weyl group B_31.

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A161987 Number of reduced words of length n in the Weyl group B_32.

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A161988 Number of reduced words of length n in the Weyl group B_33.

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