A007851 Number of elements w of the Weyl group D(n) such that the roots sent negative by w span an Abelian subalgebra of the Lie algebra.

1, 4, 14, 48, 167, 593, 2144, 7864, 29171, 109173, 411501, 1560089, 5943199, 22732739, 87253604, 335897864, 1296447899, 5015206349, 19439895089, 75487384829, 293595204239, 1143532045499, 4459774977449, 17413705988873

Offset: 1

Author

C. Kenneth Fan [ ckfan(AT)MIT.EDU ]

Keywords

• nonn
• easy

Links

• Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 FC Type D.
• C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.
• C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
• J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.

Programs

Mathematica

Table[(n+3) CatalanNumber[n]/2-1,{n,30}] (* Harvey P. Dale, Oct 06 2017 *)

Formula

a(n) = (n+3)*C(n)/2 - 1, where C(n) is a Catalan number (see A000108).

D-finite with recurrence: -(n+1)*(3*n^2+n-12)*a(n) +(15*n^3+14*n^2-85*n+36)*a(n-1) -2*(2*n-3)*(3*n^2+7*n-8)*a(n-2)=0. - R. J. Mathar, Jun 11 2019