Erik Insko has authored 3 sequences.
A234598
Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra of so(2n).
Original entry on oeis.org
9, 18, 35, 82, 180, 385, 846, 1853, 4034, 8810, 19249, 42014, 91727, 200298, 437316, 954809, 2084746, 4551801, 9938290, 21699138, 47377577, 103443386, 225856667, 493131922, 1076696324, 2350841633, 5132790390, 11206852917, 24468864530
Offset: 4
For n = 8, a(n) = 107+73 = 180 and a(n) = 3(34) + 2(14) + 6(7) + 2(4) = 180.
- P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
-
r:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 4:
elif n=5 then return 7:
elif n=6 then return 14:
elif n=7 then return 34:
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<=3 then return 0:
elif n=4 then return 9:
elif n=5 then return 18:
elif n=6 then return 35:
elif n=5 then return 82:
else return
3*r(n-1)+2*r(n-2)+6*r(n-3)+2*r(n-4):
end if;
end proc:
-
LinearRecurrence[{1, 1, 3, 1}, {9, 18, 35, 82}, 30] (* Jean-François Alcover, Dec 06 2017 *)
A234597
Number of Weyl group elements, not containing an s_1 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 D and rank n.
Original entry on oeis.org
5, 11, 21, 48, 107, 229, 501, 1099, 2394, 5225, 11417, 24923, 54409, 118808, 259403, 566361, 1236597, 2699975, 5895058, 12871185, 28102765, 61359099, 133970477, 292509056, 638659595, 1394439181, 3044596421, 6647523443, 14514097002, 31689848889, 69191112641
Offset: 4
For n=6, a(6) = A234576(6) + A234576(5)= 14+7 = 21.
- P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
- 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. USA, 44 (No. 6, June 1958), 588-589.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
-
r:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 4:
elif n=5 then return 7:
elif n=6 then return 14:
elif n=7 then return 34:
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<=3 then return 0:
elif n=4 then return 5:
elif n=5 then return 11:
else return
r(n)+r(n-1):
end if;
end proc:
-
LinearRecurrence[{1,1,3,1},{5,11,21,48},40] (* Harvey P. Dale, Feb 17 2016 *)
-
Vec(-x^4*(x^3+5*x^2+6*x+5)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 30 2013
A234576
Number of Weyl group elements, not containing s_1 or s_2, 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 D and rank n.
Original entry on oeis.org
4, 7, 14, 34, 73, 156, 345, 754, 1640, 3585, 7832, 17091, 37318, 81490, 177913, 388448, 848149, 1851826, 4043232, 8827953, 19274812, 42084287, 91886190, 200622866, 438036729, 956402452, 2088193969, 4559329474, 9954767528, 21735081361, 47456031280
Offset: 4
For n = 8, a(8) = 34+14+3*7+4 = 73.
- P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
-
a:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 4:
elif n=5 then return 7:
elif n=6 then return 14:
elif n=7 then return 34:
else return
a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
end if;
end proc:
-
LinearRecurrence[{1, 1, 3, 1}, {4, 7, 14, 34}, 31] (* Jean-François Alcover, Nov 26 2017 *)
-
Vec(-x^4*(x^3+3*x^2+3*x+4)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 30 2013
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