A162297 -id:A162297 - cp's OEIS frontend

A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).

1, 3, 5, 6, 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, 0, 0, 0

Offset: 0

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

a(n) is also the number of vertices of a truncated octahedron (the Voronoi cell for the lattice A_3*) at edge distance n from a given vertex. See also row 4 of the triangle in A008302. - N. J. A. Sloane, Oct 12 2015, corrected Aug 26 2016.
If the zeros are omitted, this is the coordination sequence for the truncated octahedron (see Karzes link). - N. J. A. Sloane, Jan 08 2020
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.

Tom Karzes, Polyhedron Coordination Sequences
Index entries for growth series for groups

Cf. A008302, A161409.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) / (1 - x)^3, {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.

A162206 Triangle read by rows in which row n (n >= 1) gives coefficients in expansion of the polynomial f(n) * Product_{i=1..n-1} f(2i), where f(k) = (1 - x^k)/(1-x).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1, 1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 314, 478, 679, 908, 1151, 1390, 1605, 1776, 1886, 1924, 1886, 1776

Offset: 1

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

For n >= 3, this polynomial is the Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) D_n.
Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ...: see A002061. - Michel Marcus, May 17 2013
The asymptotic growth of maximum elements for the reflection group D_n is about 2(n-1/2) (compare with A000140). - Mikhail Gaichenkov, Aug 21 2019
Row maxima ~ 2^(n-1)*n!/(sigma * sqrt(3/Pi)), sigma^2 = (4*n^3 - 3*n^2 - n)/36 = variance of D_n. - Mikhail Gaichenkov, Feb 08 2023

			Triangle begins:
1;
1, 2, 1;
1, 3, 5, 6, 5, 3, 1;
1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1;
1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1;
1, 6, 20, 50, 104, 190, 314, 478, 679, 908, 1151, 1390, 1605, 1776, 1886, 1924, 1886, 1776, 1605, 1390, 1151, 908, 679, 478, 314, 190, 104, 50, 20, 6, 1;
1, 7, 27, 77, 181, 371, 686, 1169, 1862, 2800, 4005, 5481, 7210, 9149, 11230, 13363, 15442, 17353, 18983, 20230, 21013, 21280, 21013, 20230, 18983, 17353, 15442, 13363, 11230, 9149, 7210, 5481, 4005, 2800, 1862, 1169, 686, 371, 181, 77, 27, 7, 1;

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Jean-François Alcover, Table of n, a(n) for n = 1..9020 [30 rows]
Marwa Ben Abdelmaksoud and Adel Hamdi, width-k Eulerian polynomials of type A and B and its Gamma-positivity, arXiv:1912.08551 [math.CO], 2019.
M. Gaichenkov, The growth of maximum elements for the reflection group $D_n$, MathOverflow, 2019.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
M. Rubey, St001443: Finite Cartan types ⟶ ℤ, StatisticsDatabase, 2019.
Index entries for growth series for groups

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Cf. A002061 (row length), A002866 (row sum), A333508 (central coefficient).

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

T[nn_] := Reap[Do[x = y + y O[y]^(n^2); v = (1 - x^n) Product[1 - x^(2k), {k, 1, n - 1}]/(1 - x)^n // CoefficientList[#, y]&; Sow[v], {n, nn}]][[2, 1]];
T[6] // Flatten (* Jean-François Alcover, Mar 25 2020, after PARI *)
T[ n_] := Module[{x}, CoefficientList[ Product[1 - x^(2 k), {k, 1, n - 1}] (1 - x^n) /(1 - x)^n // Expand, x]] (* Michael Somos, Aug 06 2021 *)

{row(n) = Vec(prod(k=1 ,n-1, 1-x^(2*k))*(1-x^n)/(1-x)^n)}; /* Michael Somos, Aug 06 2021 */

Revised by N. J. A. Sloane, Jan 10 2016

A162212 Number of reduced words of length n in the Weyl group D_9.

Original entry on oeis.org

1, 9, 44, 156, 449, 1113, 2463, 4983, 9372, 16587, 27877, 44802, 69231, 103314, 149425, 210075, 287796, 384999, 503812, 645906, 812319, 1003290, 1218116, 1455045, 1711216, 1982655, 2264333, 2550288, 2833809, 3107676, 3364445, 3596763, 3797695, 3961044, 4081645

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..72
Index entries for growth series for groups

Row 9 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

A162212g := proc(m::integer)
    (1-x^m)/(1-x) ;
end proc:
A162212 := proc(n,k)
    g := A162212g(k);
    for m from 2 to 2*k-2 by 2 do
        g := g*A162212g(m) ;
    end do:
    g := expand(g) ;
    coeftayl(g,x=0,n) ;
end proc:
seq( A162212(n,9),n=0..30) ; # R. J. Mathar, Jan 19 2016
# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

n = 9;
x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162248 Number of reduced words of length n in the Weyl group D_10.

Original entry on oeis.org

1, 10, 54, 210, 659, 1772, 4235, 9218, 18590, 35178, 63063, 107900, 177243, 280850, 430939, 642364, 932680, 1322068, 1833095, 2490290, 3319525, 4347200, 5599243, 7099950, 8870703, 10928616, 13285169, 15944898, 18904214, 22150426, 25661040, 29403398, 33334708, 37402498

Offset: 0

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

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..90
Index entries for growth series for groups

Row 10 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

Entry revised by N. J. A. Sloane, Jan 17 2016

Data corrected by Jean-François Alcover, Mar 25 2020

A162288 Number of reduced words of length n in the Weyl group D_11.

Original entry on oeis.org

1, 11, 65, 275, 934, 2706, 6941, 16159, 34749, 69927, 132991, 240900, 418187, 699193, 1130581, 1774058, 2709201, 4036252, 5878719, 8385597, 11733007, 16125043, 21793619, 28997122, 38017704, 49157086, 62730799, 79060850, 98466873, 121255904, 147711001, 178079011

Offset: 0

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

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..110
Index entries for growth series for groups

Row 11 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)
x =.; n = 11; CoefficientList[ Product[1 - x^(2 k), {k, 1, n - 1}] (1 - x^n) /(1 - x)^n // Expand, x] (* Michael Somos, Aug 06 2021 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

Entry revised by N. J. A. Sloane, Jan 17 2016

A162300 Number of reduced words of length n in the Weyl group D_13.

Original entry on oeis.org

1, 13, 90, 442, 1728, 5720, 16653, 43745, 105586, 237354, 502113, 1007773, 1931631, 3554746, 6308706, 10837593, 18078112, 29360890, 46535840, 72124195, 109499325, 163097740, 238660747, 343506072, 486827392, 680018170, 937014482, 1274649714

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..156
Index entries for growth series for groups

Row 13 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

n = 13;
x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162301 Number of reduced words of length n in the Weyl group D_14.

Original entry on oeis.org

1, 14, 104, 546, 2274, 7994, 24647, 68392, 173978, 411332, 913445, 1921218, 3852849, 7407596, 13716314, 24553984, 42632448, 71994624, 118534456, 190669584, 300196001, 463355582, 702148097, 1045918928, 1533251980, 2214194008, 3152831605, 4430235278, 6147776297

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..182
Index entries for growth series for groups

Row 14 of A162206

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

n = 14;
x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162321 Number of reduced words of length n in the Weyl group D_15.

Original entry on oeis.org

1, 15, 119, 665, 2939, 10933, 35580, 103972, 277950, 689282, 1602727, 3523945, 7376794, 14784390, 28500705, 53054702, 95687240, 167682306, 286218490, 476893794, 777106448, 1240505775, 1942759458, 2988915740, 4522669833, 6737871614, 9892634850, 14326424875

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..210
Index entries for growth series for groups

Row 15 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

n = 15;
x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162327 Number of reduced words of length n in the Weyl group D_16.

Original entry on oeis.org

1, 16, 135, 800, 3739, 14672, 50252, 154224, 432174, 1121456, 2724183, 6248128, 13624922, 28409312, 56910017, 109964720, 205651974, 373334384, 659553420, 1136449488, 1913563930, 3154094352, 5096922202, 8086011920, 12609093085, 19347878144, 29242434212, 43572711936

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..240
Index entries for growth series for groups

Row 16 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162328 Number of reduced words of length n in the Weyl group D_17.

Original entry on oeis.org

1, 17, 152, 952, 4691, 19363, 69615, 223839, 656013, 1777469, 4501652, 10749780, 24374702, 52784014, 109694031, 219658751, 425310726, 798645125, 1458198664, 2594648817, 4508215686, 7662320971, 12759278753, 20845394645, 33454765680, 52803333106, 82047370045

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 IV.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Andrew Howroyd, Table of n, a(n) for n = 0..272
Index entries for growth series for groups

Row 17 of A162206.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

cp's OEIS Frontend

A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A162206 Triangle read by rows in which row n (n >= 1) gives coefficients in expansion of the polynomial f(n) * Product_{i=1..n-1} f(2i), where f(k) = (1 - x^k)/(1-x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A162212 Number of reduced words of length n in the Weyl group D_9.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A162248 Number of reduced words of length n in the Weyl group D_10.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A162288 Number of reduced words of length n in the Weyl group D_11.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A162300 Number of reduced words of length n in the Weyl group D_13.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A162301 Number of reduced words of length n in the Weyl group D_14.

Views

Author

Keywords

References

Links

Crossrefs