A161409 -id:A161409 - cp's OEIS frontend

A162497 Number of reduced words of length n in the reflection group [3,3,5] of order 14400.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 168, 192, 216, 240, 264, 288, 312, 336, 359, 380, 399, 416, 431, 444, 455, 464, 471, 476, 478, 476, 471, 464, 455, 444, 431, 416, 399, 380, 359, 336, 312, 288, 264, 240, 216, 192, 168, 144, 121, 100, 81, 64, 49, 36, 25, 16

Offset: 0

John Cannon and N. J. A. Sloane, Dec 01 2009

This is also the Weyl group H_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-A162496.

G := CoxeterGroup(GrpFPCox, "H4");
f := GrowthFunction(G);
Coefficients(f);

G.f.: (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/(1-x)^4.

A161410 Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

Original entry on oeis.org

1, 7, 27, 77, 183, 385, 740, 1325, 2242, 3623, 5633, 8474, 12391, 17676, 24670, 33768, 45426, 60164, 78568, 101296, 129083, 162742, 203168, 251346, 308355, 375369, 453663, 544620, 649732, 770602, 908952, 1066628, 1245600, 1447967, 1675965, 1931969, 2218494

Offset: 0

John Cannon and N. J. A. Sloane, Nov 29 2009

			Coxeter matrix:
. [1 2 3 2 2 2 2]
. [2 1 2 3 2 2 3]
. [3 2 1 3 2 2 2]
. [2 3 3 1 3 2 2]
. [2 2 2 3 1 3 2]
. [2 2 2 2 3 1 2]
. [2 3 2 2 2 2 1]

N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A161409, A154638.

Z := Integers();
C := SymmetricMatrix(
[1,
2,1,
3,2,1,
2,3,3,1,
2,2,2,3,1,
2,2,2,2,3,1,
2,3,2,2,2,2,1]);
G := CoxeterGroup(GrpFPCox, C);
f := GrowthFunction(G);
T := PowerSeriesRing(Z, 50);
Eltseq(T!f);
// Corrected by Klaus Brockhaus, Feb 12 2010

CoefficientList[Series[(x^22 + 3 x^21 + 5 x^20 + 7 x^19 + 10 x^18 + 14 x^17 + 17 x^16 + 19 x^15 + 22 x^14 + 25 x^13 + 26 x^12 + 26 x^11 + 26 x^10 + 25 x^9 + 22 x^8 + 19 x^7 + 17 x^6 + 14 x^5 + 10 x^4 + 7 x^3 + 5 x^2 + 3 x + 1) / (x^22 - 4 x^21 + 6 x^20 - 4 x^19 + x^18 - x^15 + 4 x^14 - 6 x^13 + 4 x^12 - 2 x^11 + 4 x^10 - 6 x^9 + 4 x^8 - x^7 + x^4 - 4 x^3 + 6 x^2 - 4 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

G.f.: (x^22 + 3*x^21 + 5*x^20 + 7*x^19 + 10*x^18 + 14*x^17 + 17*x^16 + 19*x^15 + 22*x^14 + 25*x^13 + 26*x^12 + 26*x^11 + 26*x^10 + 25*x^9 + 22*x^8 + 19*x^7 + 17*x^6 + 14*x^5 + 10*x^4 + 7*x^3 + 5*x^2 + 3*x + 1)/(x^22 - 4*x^21 + 6*x^20 - 4*x^19 + x^18 - x^15 + 4*x^14 - 6*x^13 + 4*x^12 - 2*x^11 + 4*x^10 - 6*x^9 + 4*x^8 - x^7 + x^4 - 4*x^3 + 6*x^2 - 4*x + 1)

A161879 Number of reduced words of length n in the Weyl group B_19.

Original entry on oeis.org

1, 19, 189, 1311, 7124, 32300, 127091, 445721, 1420364, 4172476, 11426240, 29429784, 71808030, 166970290, 371826581, 796341623, 1646167391, 3294638285, 6401307860, 12102626404, 22312161586, 40184022430, 70815181390, 122291804610, 207223417349, 344959019207, 564743768579

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..361

Row n=19 of A128084.

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.

A161900 Number of reduced words of length n in the Weyl group B_22.

Original entry on oeis.org

1, 22, 252, 2002, 12396, 63734, 283107, 1116236, 3983485, 13057330, 39764011, 113533312, 306173263, 784654154, 1920802566, 4510960122, 10201294213, 22286443124, 47167714715, 96947735390, 193938666735, 378324531180, 720920510115, 1344018408150, 2454841642634

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..484

Row n=22 of A128084.

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.

A161930 Number of reduced words of length n in the Weyl group B_23.

Original entry on oeis.org

1, 23, 275, 2277, 14673, 78407, 361514, 1477750, 5461235, 18518565, 58282576, 171815888, 477989151, 1262643305, 3183445871, 7694405993, 17895700206, 40182143330, 87349858045, 184297593435, 378236260170, 756560791350, 1477481301465, 2821499709615, 5276341352249

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..529

Row n=23 of A128084.

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.

A161931 Number of reduced words of length n in the Weyl group B_24.

Original entry on oeis.org

1, 24, 299, 2576, 17249, 95656, 457170, 1934920, 7396155, 25914720, 84197296, 256013184, 734002335, 1996645640, 5180091511, 12874497504, 30770197710, 70952341040, 158302199085, 342599792520, 720836052690, 1477396844040, 2954878145505, 5776377855120, 11052719207369

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..576

Row n=24 of A128084.

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.

A161932 Number of reduced words of length n in the Weyl group B_25.

Original entry on oeis.org

1, 25, 324, 2900, 20149, 115805, 572975, 2507895, 9904050, 35818770, 120016066, 376029250, 1110031585, 3106677225, 8286768736, 21161266240, 51931463950, 122883804990, 281186004075, 623785796595, 1344621849285, 2822018693325, 5776896838830, 11553274693950, 22605993901319

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..625

Row n=25 of A128084.

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.

A161933 Number of reduced words of length n in the Weyl group B_26.

Original entry on oeis.org

1, 26, 350, 3250, 23399, 139204, 712179, 3220074, 13124124, 48942894, 168958960, 544988210, 1655019795, 4761697020, 13048465756, 34209731996, 86141195946, 209025000936, 490211005011, 1113996801606, 2458618650891, 5280637344216, 11057534183046, 22610808876996

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..676

Row n=26 of A128084.

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.

A161436 Number of reduced words of length n in the Weyl group A_4.

Original entry on oeis.org

1, 4, 9, 15, 20, 22, 20, 15, 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, 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.

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

Cf. A008302, A161409.

CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) / (1 - x)^4, {x, 0, 20}], x] (* Vincenzo Librandi, Aug 23 2016 *)

G.f. for A_m is the polynomial Product_{k=1..m}(1-x^(k+1))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A008302.

A161437 Number of reduced words of length n in the Weyl group A_5.

Original entry on oeis.org

1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 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

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 I.)

CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) / (1 - x)^5, {x, 0, 120}], x] (* Vincenzo Librandi, Aug 23 2016 *)

G.f. for A_m is the polynomial Prod_{k=1..m}(1-x^(k+1))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A008302.

cp's OEIS Frontend

A162497 Number of reduced words of length n in the reflection group [3,3,5] of order 14400.

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A161410 Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

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A161879 Number of reduced words of length n in the Weyl group B_19.

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A161900 Number of reduced words of length n in the Weyl group B_22.

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A161930 Number of reduced words of length n in the Weyl group B_23.

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A161931 Number of reduced words of length n in the Weyl group B_24.

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A161932 Number of reduced words of length n in the Weyl group B_25.

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A161933 Number of reduced words of length n in the Weyl group B_26.

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A161436 Number of reduced words of length n in the Weyl group A_4.

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