A243094 Cardinality of the Weyl alternation set corresponding to the zero-weight in the representation of the Lie algebra sp(2n) whose highest weight is the second fundamental weight.

1, 2, 5, 8, 19, 44, 92, 201, 444, 965, 2104, 4602, 10045, 21924, 47879, 104540, 228236, 498337, 1088072, 2375657, 5186976, 11325186, 24727205, 53988976, 117878715, 257374492, 561947340, 1226946953, 2678896484, 5849059949, 12770744632, 27883440986, 60880261949

Offset

0,2

Comments

Number of Weyl group elements contributing nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the defining representation for the Lie algebra of type C and rank n. Here the highest weight would be the second fundamental weight of sp(2n).

Links

Table of n, a(n) for n=0..32.

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

P. E. Harris, E. Insko, 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.

Index entries for linear recurrences with constant coefficients, signature (1, 1, 3, 1).

Formula

a(n) = A232162(n) + A232162(n-1).

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

G.f.: (x^4 + 2*x^3 - 2*x^2 - x - 1) / (x^4 + 3*x^3 + x^2 + x - 1). - Joerg Arndt, Aug 18 2014

Maple

r:=proc(n::nonnegint) option remember
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
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:

Mathematica

Join[{1}, LinearRecurrence[{1, 1, 3, 1}, {2, 5, 8, 19}, 32]] (* Jean-François Alcover, Dec 05 2017 *)

Prog

(PARI) Vec( (x^4+2*x^3-2*x^2-x-1) / (x^4+3*x^3+x^2+x-1) +O(x^66) ) \\ Joerg Arndt, Aug 18 2014

Crossrefs

Cf. A232162.

Sequence in context: A320295 A277263 A055614 * A076870 A127281 A054774

Adjacent sequences: A243091 A243092 A243093 * A243095 A243096 A243097

Keyword

nonn

Author

Pamela E Harris, Aug 18 2014

Status

approved