A161409 -id:A161409 - cp's OEIS frontend

A161438 Number of reduced words of length n in the Weyl group A_6.

1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 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

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.)

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.

A161439 Number of reduced words of length n in the Weyl group A_7.

Original entry on oeis.org

1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 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.

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.

Row n=8 of A008302.

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.

A161456 Number of reduced words of length n in the Weyl group A_8.

Original entry on oeis.org

1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche I.)

Row n=9 of A008302.

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.

A161457 Number of reduced words of length n in the Weyl group A_9.

Original entry on oeis.org

1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, 32683, 47043, 64889, 86054, 110010, 135853, 162337, 187959, 211089, 230131, 243694, 250749, 250749, 243694, 230131, 211089, 187959, 162337, 135853, 110010, 86054, 64889, 47043, 32683, 21670, 13640, 8095, 4489, 2298, 1068, 440, 155, 44, 9, 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.

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[QFactorial[9+1,q],{q,0,9*(9+1)/2}],q] (* Wouter Meeussen, Jul 12 2014 *)

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.

A161458 Number of reduced words of length n in the Weyl group A_10.

Original entry on oeis.org

1, 10, 54, 209, 649, 1717, 4015, 8504, 16599, 30239, 51909, 84591, 131625, 196470, 282369, 391939, 526724, 686763, 870233, 1073227, 1289718, 1511742, 1729808, 1933514, 2112319, 2256396, 2357475, 2409581, 2409581, 2357475, 2256396, 2112319, 1933514, 1729808, 1511742

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..55

Row n=11 of A008302.

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.

A161459 Number of reduced words of length n in the Weyl group A_11.

Original entry on oeis.org

1, 11, 65, 274, 923, 2640, 6655, 15159, 31758, 61997, 113906, 198497, 330121, 526581, 808896, 1200626, 1726701, 2411747, 3277965, 4342688, 5615807, 7097310, 8775209, 10624132, 12604826, 14664752, 16739858, 18757500, 20640357, 22311069, 23697232, 24736324, 25380120

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..66

Row n=12 of A008302.

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.

A161461 Number of reduced words of length n in the Weyl group A_12.

Original entry on oeis.org

1, 12, 77, 351, 1274, 3914, 10569, 25728, 57486, 119483, 233389, 431886, 762007, 1288587, 2097472, 3298033, 5024460, 7435284, 10710609, 15046642, 20647290, 27712842, 36426054, 46936280, 59342609, 73677240, 89890517, 107839121, 127278852, 147863220, 169148705

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..78

Row n=13 of A008302.

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.

A161465 Number of reduced words of length n in the Weyl group A_13.

Original entry on oeis.org

1, 13, 90, 441, 1715, 5629, 16198, 41926, 99412, 218895, 452284, 884170, 1646177, 2934764, 5032235, 8330256, 13354639, 20789572, 31498907, 46541635, 67178356, 94865470, 131234038, 178050835, 237160055, 310405409, 399533919, 506084453, 631265833, 775831020, 939955265

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..91

Row n=14 of A008302.

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.

A161475 Number of reduced words of length n in the Weyl group A_14.

Original entry on oeis.org

1, 14, 104, 545, 2260, 7889, 24087, 66013, 165425, 384320, 836604, 1720774, 3366951, 6301715, 11333950, 19664205, 33018831, 53808313, 85306779, 131846699, 199019426, 293868698, 425060810, 603012233, 839953393, 1149906518, 1548556267, 2052994543, 2681325612

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..105

Row n=15 of A008302.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
coeffs(b(15, 0));  # Alois P. Heinz, Mar 21 2025

b[u_, o_] := b[u, o] = Expand[If[u+o == 0, 1, Sum[b[u+j-1, o-j]*x^(u+j-1), {j, 1, o}] + Sum[b[u-j, o+j-1]*x^(u-j), {j, 1, u}]]];
CoefficientList[b[15, 0], x] (* Jean-François Alcover, May 16 2025, after Alois P. Heinz *)

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.

A161476 Number of reduced words of length n in the Weyl group A_15.

Original entry on oeis.org

1, 15, 119, 664, 2924, 10813, 34900, 100913, 266338, 650658, 1487262, 3208036, 6574987, 12876702, 24210652, 43874857, 76893687, 130701986, 216008661, 347854815, 546871981, 840732790, 1265769513, 1868715733, 2708503701, 3858025899, 5405745562, 7457019331, 10134977992

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 I.)

Andrew Howroyd, Table of n, a(n) for n = 0..120

Row n=16 of A008302.

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

A161438 Number of reduced words of length n in the Weyl group A_6.

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A161439 Number of reduced words of length n in the Weyl group A_7.

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A161456 Number of reduced words of length n in the Weyl group A_8.

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A161457 Number of reduced words of length n in the Weyl group A_9.

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A161458 Number of reduced words of length n in the Weyl group A_10.

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A161459 Number of reduced words of length n in the Weyl group A_11.

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A161461 Number of reduced words of length n in the Weyl group A_12.

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A161465 Number of reduced words of length n in the Weyl group A_13.

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A161475 Number of reduced words of length n in the Weyl group A_14.

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