A008137 -id:A008137 - cp's OEIS frontend

A161696 Number of reduced words of length n in the Weyl group B_3.

1, 3, 5, 7, 8, 8, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

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

nonn

If the zeros are ignored, this is the coordination sequence for the truncated cuboctahedron (see the Karzes link). - N. J. A. Sloane, Jan 08 2020

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.)

Tom Karzes, Polyhedron Coordination Sequences

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:=10; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..3),x,n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,3}] /(1-x)^3, {x,0,9}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^10); Vec(prod(k=1,3,1-t^(2*k))/(1-t)^3) \\ 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.

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

Original entry on oeis.org

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.

A267167 Growth series for affine Coxeter group B_4.

Original entry on oeis.org

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].

cp's OEIS Frontend

A161696 Number of reduced words of length n in the Weyl group B_3.

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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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