A232162 Number of Weyl group elements, not containing an s_r factor, which contribute nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the adjoint representation for the Lie algebra of type B and rank n.
0, 0, 2, 3, 5, 14, 30, 62, 139, 305, 660, 1444, 3158, 6887, 15037, 32842, 71698, 156538, 341799, 746273, 1629384, 3557592, 7767594, 16959611, 37029365, 80849350, 176525142, 385422198, 841524755, 1837371729, 4011688220, 8759056412, 19124384574, 41755877375, 91169119405
Offset: 0
Examples
For n=4, a(4) = A232162(3) + A232162(2) + 3*A232162(1) + A232162(0) = 3+2+3*0+0=5.
References
- P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
Links
- Stefano Spezia, Table of n, a(n) for n = 0..2950
- P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
- B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
- László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 1 at p. 3.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
Programs
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Maple
a:=proc(n::nonnegint) if n=0 then return 0: elif n=1 then return 0: elif n=2 then return 2: elif n=3 then return 3: else return a(n-1)+a(n-2)+3*a(n-3)+a(n-4): end if; end proc:
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Mathematica
LinearRecurrence[{1, 1, 3, 1}, {0, 0, 2, 3}, 32] (* Jean-François Alcover, Nov 24 2017 *) -
PARI
Vec(-x^2*(x+2)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013
Formula
From Colin Barker, Dec 31 2013: (Start)
a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: -x^2*(x + 2)/(x^4 + 3*x^3 + x^2 + x - 1). (End)