A232164 -id:A232164 - cp's OEIS frontend

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

Pamela E Harris, Nov 19 2013

			For n=4, a(4) = A232162(3) + A232162(2) + 3*A232162(1) + A232162(0) = 3+2+3*0+0=5.

P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

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).

Cf. A232163, A232164, A232165.

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:

LinearRecurrence[{1, 1, 3, 1}, {0, 0, 2, 3}, 32] (* Jean-François Alcover, Nov 24 2017 *)

Vec(-x^2*(x+2)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013

a(n) = A232162(n-1) + A232162(n-2) + 3*A232162(n-3) + A232162(n-4).
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)

A232165 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra sp(2n).

Original entry on oeis.org

0, 1, 2, 3, 8, 18, 37, 82, 181, 392, 856, 1873, 4086, 8919, 19480, 42530, 92853, 202742, 442665, 966496, 2110240, 4607473, 10059866, 21964555, 47957080, 104708706, 228619317, 499163818, 1089866333, 2379596808, 5195573912, 11343933537, 24768164206, 54078416287

Offset: 0

Pamela E Harris, Nov 19 2013

Number of Weyl group elements contributing 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 C and rank n.

			For n=3, a(3) = A232164(3) + A232164(2) = 2+1 = 3.

P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

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, 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).

Cf. A232164.

r:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
elif n=2 then return 1:
elif n=3 then return 2:
else return
r(n-1)+r(n-2)+3*r(n-3)+r(n-4):
end if;
end proc:
a:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
else return
r(n)+r(n-1):
end if;
end proc:

LinearRecurrence[{1,1,3,1},{0,1,2,3},40] (* Harvey P. Dale, Nov 22 2014 *)

Vec(-x*(x+1)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Jan 01 2014

a(n) = A232164(n) + A232164(n-1).
From Colin Barker, Jan 01 2014: (Start)
a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: -x*(x + 1)/(x^4 + 3*x^3 + x^2 + x - 1). (End)

cp's OEIS Frontend

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.

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