A054761 Number of positive braids with n crossings of 5 strands.

1, 4, 13, 37, 99, 254, 636, 1567, 3822, 9261, 22346, 53773, 129174, 309958, 743228, 1781330, 4268166, 10224885, 24492034, 58662298, 140498877, 336491169, 805872377, 1929983778, 4622083068, 11069289411, 26509431448, 63486333364

Offset: 0

Author

Serge Burckel (burckel(AT)univ-reunion.fr), Apr 27 2000

Keywords

• nonn

Comments

The (n+1)-strand braid group B_{n+1} has n generators s_1,...,s_n with relations s_i s_k s_i = s_k s_i s_k for k=i+1, s_i s_k = s_k s_i for k>i+1. The elements are the isotopy classes of (n+1) free strands that are planarily "mixed" (s_i corresponds to the operation of crossing the i-th strand under the (i+1)-th strand).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• F. A. Garside, The braid group and other groups, Quart. Journal Math. Oxford, Volume 20, Issue 1, 1 January 1969, Pages 235-254.
• Kyoji Saito, Growth functions associated with Artin monoids of finite type, Proc. Japan Acad. Ser. A Math. Sci., 84 (2008), no. 10, 179-183. [_Ignat Soroko_, Sep 30 2010]
• Index entries for linear recurrences with constant coefficients, signature (4, -3, -3, 2, 0, 2, 0, 0, 0, -1).

Crossrefs

Cf. A000071, A054480.

Programs

Magma

I:=[1,4,13,37,99,254,636,1567,3822,9261]; [n le 10 select I[n] else 4*Self(n-1) - 3*Self(n-2) -3*Self(n-3) +2*Self(n-4) +2*Self(n-6) -Self(n-10): n in [1..30]];

Mathematica

CoefficientList[Series[1/(1-4x+3x^2+3x^3-2x^4-2x^6+x^10),{x,0,30}],x] (* or *) LinearRecurrence[{4,-3,-3,2,0,2,0,0,0,-1}, {1,4,13,37,99,254,636, 1567,3822,9261},30] (* Harvey P. Dale, May 09 2017 *)

PARI

x='x+O('x^30); Vec(1/(1-4*x+3*x^2+3*x^3-2*x^4-2*x^6+x^10)) \\ G. C. Greubel, Jan 17 2018

Formula

G.f.: 1/(1-4*x+3*x^2+3*x^3-2*x^4-2*x^6+x^10). - Ignat Soroko, Sep 30 2010

a(n) = 4*a(n-1) - 3*a(n-2) - 3*a(n-3) + 2*a(n-4) + 2*a(n-6) - a(n-10). - Wesley Ivan Hurt, May 12 2023

Extensions

More terms from Ignat Soroko, Sep 30 2010