A008137 -id:A008137 - cp's OEIS frontend

A242941 a(n) is the number of convex uniform tessellations in dimension n.

1, 11, 28, 143

Offset: 1

Felix Fröhlich, May 27 2014

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.
Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014
From Felix Fröhlich, Feb 03 2019: (Start)
The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative [On the regular and semiregular nets of polyhedra and on the corresponding correlative nets], Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
M. Deza and M. Shtogrin, Uniform Partitions of 3-space, their Relatives and Embedding, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
B. Grünbaum and G. C. Shephard, Tilings by Regular Polygons, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
J. Kepler, Harmonices Mundi [The Harmony of the World] (1619).
G. Olshevsky, Uniform Panoploid Tetracombs (2006)
A. I. Weiss and E. M. Stehle, Norman W. Johnson (12 November 1930 to 13 July 2017), The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
Wikipedia, Convex uniform honeycomb
Wikipedia, List of convex uniform tilings

Cf. A068599.
List of coordination sequences for the 11 uniform 2D tilings: A008458(the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for the 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Edited by N. J. A. Sloane, Feb 15 2018

Edited by Felix Fröhlich, Feb 03-10 2019

A267168 Growth series for affine Coxeter group B_5.

Original entry on oeis.org

1, 6, 20, 51, 110, 211, 372, 615, 966, 1455, 2117, 2991, 4120, 5551, 7334, 9524, 12180, 15365, 19146, 23594, 28784, 34795, 41711, 49619, 58611, 68783, 80234, 93067, 107389, 123312, 140952, 160430, 181870, 205400, 231152, 259261, 289867, 323114, 359151, 398131, 440211, 485551, 534315, 586672, 642794, 702858, 767045, 835540, 908532, 986214

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 (3, -3, 1, 0, 0, 0, 1, -3, 4, -4, 3, -1, 0, 0, 0, -1, 3, -3, 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].

Here (k=5) the G.f. is -(1+t)*(1+t+t^2+t^3)*(t^3+1)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^5+1)/(-1+t^9)/(-1+t^7)/(-1+t)^3.

A267169 Growth series for affine Coxeter group B_6.

Original entry on oeis.org

1, 7, 27, 78, 188, 399, 771, 1386, 2352, 3807, 5924, 8916, 13041, 18606, 25971, 35554, 47835, 63361, 82750, 106695, 135968, 171425, 214011, 264764, 324820, 395417, 477900, 573724, 684459, 811795, 957546, 1123655, 1312198, 1525389, 1765583, 2035281, 2337134, 2673948, 3048689, 3464488, 3924646, 4432636, 4992108, 5606893, 6281008, 7018660

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 (4, -6, 3, 3, -6, 3, 4, -10, 10, -4, -2, 2, 3, -6, 3, 2, -2, -4, 10, -10, 4, 3, -6, 3, 3, -6, 4, -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].

A267170 Growth series for affine Coxeter group B_7.

Original entry on oeis.org

1, 8, 35, 113, 301, 700, 1471, 2857, 5209, 9016, 14940, 23856, 36897, 55504, 81481, 117055, 164941, 228412, 311373, 418440, 555023, 727414, 942880, 1209761, 1537573, 1937115, 2420581, 3001676, 3695738, 4519865, 5493047, 6636302, 7972817, 9528094, 11330100, 13409422, 15799426, 18536422, 21659833, 25212370, 29240211, 33793185, 38924961

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 (5, -10, 9, 0, -9, 9, 0, -9, 10, -5, 2, -5, 11, -14, 10, 0, -9, 9, 0, -10, 14, -11, 5, -2, 5, -10, 9, 0, -9, 9, 0, -9, 10, -5, 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].

A267171 Growth series for affine Coxeter group B_8.

Original entry on oeis.org

1, 9, 44, 157, 458, 1158, 2629, 5486, 10695, 19711, 34651, 58507, 95404, 150908, 232389, 349445, 514393, 742832, 1054283, 1472911, 2028333, 2756518, 3700784, 4912897, 6454277, 8397316, 10826813, 13841530, 17555875, 22101717, 27630339, 34314534, 42350849, 51961982, 63399337, 76945741, 92918329, 111671603, 133600669, 159144658, 188790335

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 (5, -10, 9, 0, -9, 9, 0, -9, 10, -5, 2, -5, 11, -14, 11, -5, 1, 0, 0, -1, 5, -11, 14, -12, 10, -12, 14, -11, 5, -1, 0, 0, 1, -5, 11, -14, 11, -5, 2, -5, 10, -9, 0, 9, -9, 0, 9, -10, 5, -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].

A267172 Growth series for affine Coxeter group B_9.

Original entry on oeis.org

1, 10, 54, 211, 669, 1827, 4456, 9942, 20637, 40348, 74999, 133506, 228910, 379818, 612207, 961652, 1476045, 2218878, 3273169, 4746115, 6774561, 9531380, 13232864, 18147232, 24604366, 33006891, 43842720, 57699190, 75278921, 97417535, 125103378, 159499393, 201967298, 254094228, 317722005, 394979205, 488316197, 600543335, 734872490

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 (6, -15, 19, -9, -9, 18, -9, -9, 18, -9, -8, 12, 7, -34, 43, -25, -2, 11, 6, -28, 27, 0, -27, 26, 6, -43, 52, -26, -5, 5, 26, -52, 43, -6, -26, 27, 0, -27, 28, -6, -11, 2, 25, -43, 34, -7, -12, 8, 9, -18, 9, 9, -18, 9, 9, -19, 15, -6, 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].

A267173 Growth series for affine Coxeter group B_10.

Original entry on oeis.org

1, 11, 65, 276, 945, 2772, 7228, 17170, 37807, 78155, 153154, 286660, 515570, 895388, 1507595, 2469247, 3945292, 6164170, 9437339, 14183455, 20958025, 30489449, 43722470, 61870160, 86475684, 119485204, 163333410, 221043295, 296341927, 393794113, 518955998, 678550795, 880669001, 1134995618, 1453067068, 1848560666, 2337619696, 2939217322

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 (7, -21, 34, -28, -1, 34, -48, 34, -1, -28, 35, -28, 29, -42, 49, -33, -5, 42, -46, 8, 43, -66, 43, 7, -39, 21, 29, -62, 55, -30, 22, -44, 74, -71, 18, 52, -84, 52, 18, -71, 74, -44, 22, -30, 55, -62, 29, 21, -39, 7, 43, -66, 43, 8, -46, 42, -5, -33, 49, -42, 29, -28, 35, -28, -1, 34, -48, 34, -1, -28, 34, -21, 7, -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].

A267174 Growth series for affine Coxeter group B_11.

Original entry on oeis.org

1, 12, 77, 353, 1298, 4070, 11298, 28468, 66275, 144430, 297584, 584244, 1099814, 1995202, 3502797, 5972044, 9917336, 16081506, 25518845, 39702300, 60660325, 91149775, 134872255, 196742469, 283218364, 402704237, 566039474, 787087225, 1083439094, 1477253844, 1996250190, 2674875984, 3555678491, 4690903019, 6144349905, 7993522778

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 (8, -28, 55, -62, 27, 35, -82, 82, -35, -27, 62, -55, 29, -16, 29, -55, 63, -35, -6, 19, 9, -55, 82, -62, 0, 62, -82, 54, 0, -56, 98, -120, 126, -120, 98, -57, 8, 26, -27, -1, 35, -55, 55, -35, 1, 27, -26, -8, 57, -98, 120, -126, 120, -98, 56, 0, -54, 82, -62, 0, 62, -82, 55, -9, -19, 6, 35, -63, 55, -29, 16, -29, 55, -62, 27, 35, -82, 82, -35, -27, 62, -55, 28, -8, 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].

A161697 Number of reduced words of length n in the Weyl group B_4.

Original entry on oeis.org

1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 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, 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 algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

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:=17; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..4]])/(1-t)^4)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..4),x,n+1), x, n), n = 0 .. 100); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,4}] /(1-x)^4, {x,0,16}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^17); Vec(prod(k=1,4,1-t^(2*k))/(1-t)^4) \\ 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.

A161698 Number of reduced words of length n in the Weyl group B_5.

Original entry on oeis.org

1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 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.)

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:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..5]])/(1-t)^5)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..5),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,5}] /(1-x)^5, {x,0,25}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^26); Vec(prod(k=1,5,1-t^(2*k))/(1-t)^5) \\ 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.

cp's OEIS Frontend

A242941 a(n) is the number of convex uniform tessellations in dimension n.

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A267168 Growth series for affine Coxeter group B_5.

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A267169 Growth series for affine Coxeter group B_6.

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A267170 Growth series for affine Coxeter group B_7.

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A267171 Growth series for affine Coxeter group B_8.

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A267172 Growth series for affine Coxeter group B_9.

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A267173 Growth series for affine Coxeter group B_10.

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A267174 Growth series for affine Coxeter group B_11.

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A161697 Number of reduced words of length n in the Weyl group B_4.

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A161698 Number of reduced words of length n in the Weyl group B_5.