A161699 -id:A161699 - cp's OEIS frontend

A267167 Growth series for affine Coxeter group B_4.

1, 5, 14, 31, 59, 101, 161, 243, 351, 488, 658, 865, 1112, 1403, 1741, 2130, 2574, 3077, 3643, 4274, 4974, 5747, 6597, 7528, 8543, 9646, 10840, 12129, 13517, 15007, 16603, 18309, 20129, 22066, 24123, 26304, 28613, 31054, 33631, 36347, 39205, 42209, 45363, 48671, 52136, 55762, 59553, 63512, 67643, 71949, 76434, 81102, 85957, 91003, 96242

Offset: 0

N. J. A. Sloane, Jan 11 2016

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).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,2,-2,1,0,0,-1,2,-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.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7)))); // G. C. Greubel, Oct 24 2018

seq(coeff(series((1-x^2)*(1+x^3)*(1-x^4)*(1-x^8)/((1-x)^5*(1-x^5)*(1-x^7)),x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7)), {t, 0, 50}], t] (* G. C. Greubel, Oct 24 2018 *)

t='t+O('t^40); Vec((1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7))) \\ G. C. Greubel, Oct 24 2018

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=4) the G.f. is (1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^3+1)*(1+t+t^2+t^3)*(1+t) / (-1+t^7)/(-1+t^5)/(-1+t)^2.
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9) - a(n-12) + 2*a(n-13) - a(n-14), n > 0. - Muniru A Asiru, Oct 25 2018

A267175 Growth series for affine Coxeter group B_12.

Original entry on oeis.org

1, 13, 90, 443, 1741, 5811, 17109, 45577, 111852, 256282, 553866, 1138110, 2237924, 4233126, 7735923, 13707967, 23625303, 39706809, 65225654, 104927954, 165588279, 256738054, 391610309, 588352779, 871571154, 1274275456, 1840315206, 2627403376, 3710845242, 5188106314, 7184373674, 9859287465, 13415044111, 18106100284, 24250736849

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 (9, -36, 82, -108, 53, 90, -225, 217, -27, -217, 306, -144, -133, 261, -99, -217, 379, -197, -188, 404, -207, -252, 542, -360, -154, 521, -387, -107, 459, -326, -117, 358, -88, -452, 693, -327, -342, 666, -296, -434, 800, -415, -349, 720, -315, -460, 811, -359, -457, 800, -277, -649, 1102, -649, -277, 800, -457, -359, 811, -460, -315, 720, -349, -415, 800, -434, -296, 666, -342, -327, 693, -452, -88, 358, -117, -326, 459, -107, -387, 521, -154, -360, 542, -252, -207, 404, -188, -197, 379, -217, -99, 261, -133, -144, 306, -217, -27, 217, -225, 90, 53, -108, 82, -36, 9, -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].

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

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 (3, -3, 1, 0, 0, 0, 1, -3, 4, -4, 3, -1, 0, 0, 0, -1, 3, -3, 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].

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

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 (4, -6, 3, 3, -6, 3, 4, -10, 10, -4, -2, 2, 3, -6, 3, 2, -2, -4, 10, -10, 4, 3, -6, 3, 3, -6, 4, -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].

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

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, 10, 0, -9, 9, 0, -10, 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].

A267171 Growth series for affine Coxeter group B_8.

Original entry on oeis.org

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.

cp's OEIS Frontend

A267167 Growth series for affine Coxeter group B_4.

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A267175 Growth series for affine Coxeter group B_12.

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A267168 Growth series for affine Coxeter group B_5.

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A267169 Growth series for affine Coxeter group B_6.

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A267170 Growth series for affine Coxeter group B_7.

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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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