A161409 -id:A161409 - cp's OEIS frontend

A161699 Number of reduced words of length n in the Weyl group B_6.

1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919, 2254, 2565, 2832, 3037, 3166, 3210, 3166, 3037, 2832, 2565, 2254, 1919, 1580, 1255, 958, 699, 484, 315, 190, 104, 50, 20, 6, 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.)

G. C. Greubel, Table of n, a(n) for n = 0..36

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*k): k in [1..6]])/(1-t)^6)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..6),x,n+1), x, n), n = 0 .. 36); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) / (1 - x)^6, {x, 0, 50}], x]  (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,6,1-t^(2*k))/(1-t)^6) \\ G. C. Greubel, Oct 25 2018

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

A161716 Number of reduced words of length n in the Weyl group B_7.

Original entry on oeis.org

1, 7, 27, 77, 181, 371, 686, 1170, 1869, 2827, 4082, 5662, 7581, 9835, 12399, 15225, 18242, 21358, 24464, 27440, 30162, 32510, 34376, 35672, 36336, 36336, 35672, 34376, 32510, 30162, 27440, 24464, 21358, 18242, 15225, 12399, 9835, 7581, 5662

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.)

G. C. Greubel, Table of n, a(n) for n = 0..49

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:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..7]])/(1-t)^7)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..7),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) / (1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^50); Vec(prod(k=1,7,1-t^(2*k))/(1-t)^7) \\ G. C. Greubel, Oct 25 2018

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

A161717 Number of reduced words of length n in the Weyl group B_8.

Original entry on oeis.org

1, 8, 35, 112, 293, 664, 1350, 2520, 4389, 7216, 11298, 16960, 24541, 34376, 46775, 62000, 80241, 101592, 126029, 153392, 183373, 215512, 249202, 283704, 318171, 351680, 383270, 411984, 436913, 457240, 472281, 481520, 484636, 481520, 472281

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.)

Vincenzo Librandi, Table of n, a(n) for n = 0..64

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*k): k in [1..8]])/(1-t)^8)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..8),x,65), x, n), n = 0 .. 64); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) / (1 - x)^8, {x, 0, 70}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,8,1-t^(2*k))/(1-t)^8) \\ G. C. Greubel, Oct 25 2018

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

A161733 Number of reduced words of length n in the Weyl group B_9.

Original entry on oeis.org

1, 9, 44, 156, 449, 1113, 2463, 4983, 9372, 16588, 27886, 44846, 69387, 103763, 150538, 212538, 292779, 394371, 520399, 673783, 857121, 1072521, 1321430, 1604470, 1921291, 2270451, 2649332, 3054100, 3479715, 3919995, 4367735

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.)

G. C. Greubel, Table of n, a(n) for n = 0..81

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*k): k in [1..9]])/(1-t)^9)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..9),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 -x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) / (1 - x)^9, {x, 0, 81}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,9,1-t^(2*k))/(1-t)^9) \\ G. C. Greubel, Oct 25 2018

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

A161755 Number of reduced words of length n in the Weyl group B_10.

Original entry on oeis.org

1, 10, 54, 210, 659, 1772, 4235, 9218, 18590, 35178, 63064, 107910, 177297, 281060, 431598, 644136, 936915, 1331286, 1851685, 2525468, 3382588, 4455100, 5776486, 7380800, 9301642, 11570980, 14217849, 17266966, 20737309, 24640716, 28980565

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.)

G. C. Greubel, Table of n, a(n) for n = 0..100

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*k): k in [1..10]])/(1-t)^10)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..10),x,101), x, n), n = 0 .. 100); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) (1 - x^20) / (1 - x)^10, {x, 0, 100}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,10,1-t^(2*k))/(1-t)^10) \\ G. C. Greubel, Oct 25 2018

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

A161776 Number of reduced words of length n in the Weyl group B_11.

Original entry on oeis.org

1, 11, 65, 275, 934, 2706, 6941, 16159, 34749, 69927, 132991, 240901, 418198, 699258, 1130856, 1774992, 2711907, 4043193, 5894878, 8420346, 11802934, 16258034, 22034519, 29415309, 38716897, 50287667, 64504857, 81770051

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.)

G. C. Greubel, Table of n, a(n) for n = 0..121

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*k): k in [1..11]])/(1-t)^11)); // G. C. Greubel, Oct 24 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..11),x,122), x, n), n = 0 .. 121); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[((1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) (1 - x^20) (1 - x^22)) / (1 - x)^11, {x, 0, 121}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,11,1-t^(2*k))/(1-t)^11) \\ G. C. Greubel, Oct 24 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.

A161858 Number of reduced words of length n in the Weyl group B_12.

Original entry on oeis.org

1, 12, 77, 352, 1286, 3992, 10933, 27092, 61841, 131768, 264759, 505660, 923858, 1623116, 2753972, 4528964, 7240871, 11284064, 17178942, 25599288, 37402222, 53660256, 75694775, 105110084, 143826980, 194114636, 258619428, 340389204, 442891395, 570023312, 726112969

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.)

G. C. Greubel, Table of n, a(n) for n = 0..144

Row n=12 of A128084.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..12]])/(1-t)^12)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..12),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)),{k,1,12}]/(1-x)^12,{x,0,50}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^50); Vec(prod(k=1,12,1-t^(2*k))/(1-t)^12) \\ G. C. Greubel, Oct 25 2018

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

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

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 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.

Jinyuan Wang, Table of n, a(n) for n = 0..63

Cf. A161409, A162494.

G := CoxeterGroup(GrpFPCox, "E7");
f := GrowthFunction(G);
Coefficients(f);

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

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.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A161699 Number of reduced words of length n in the Weyl group B_6.

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A161716 Number of reduced words of length n in the Weyl group B_7.

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A161717 Number of reduced words of length n in the Weyl group B_8.

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A161733 Number of reduced words of length n in the Weyl group B_9.

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A161755 Number of reduced words of length n in the Weyl group B_10.

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A161776 Number of reduced words of length n in the Weyl group B_11.

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