Pamela E Harris - cp's OEIS frontend

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

Pamela E Harris, Aug 18 2014

nonn

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).

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).

Cf. A232162.

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:

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

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

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

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)

A232164 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 C and rank n.

Original entry on oeis.org

0, 1, 1, 2, 6, 12, 25, 57, 124, 268, 588, 1285, 2801, 6118, 13362, 29168, 63685, 139057, 303608, 662888, 1447352, 3160121, 6899745, 15064810, 32892270, 71816436, 156802881, 342360937, 747505396, 1632091412, 3563482500, 7780451037, 16987713169, 37090703118

Offset: 0

Pamela E Harris, Nov 19 2013

Apart from the offset the same as A214663. - R. J. Mathar, Nov 27 2013

Apart from the initial 0, number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the last element. - Sergey Kitaev, Dec 08 2020

			For n=4, a(4)= A232164(3) + A232164(2) + 3*A232164(1) + A232164(0) = 2+1+3*1+0=6.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
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.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

a:=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
a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
end if;
end proc:

CoefficientList[Series[x/(1 - x - x^2 -3 x^3- x^4),{x, 0, 30}], x] (* Vincenzo Librandi, Dec 31 2013 *)

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

a(n) = A232164(n-1) + A232164(n-2) + 3*A232164(n-3) + A232164(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/(x^4 + 3*x^3 + x^2 + x - 1). (End)

A232163 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra so(2n+1).

Original entry on oeis.org

0, 1, 2, 5, 10, 22, 49, 106, 231, 506, 1104, 2409, 5262, 11489, 25082, 54766, 119577, 261078, 570035, 1244610, 2717456, 5933249, 12954570, 28284797, 61756570, 134838326, 294403857, 642796690, 1403472095, 3064318682, 6690584704

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 B and rank n.

			For n=8, a(8) = A232162(8) + A232162(7) + A232162(6) = 139+62+30 = 231.

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).

Cf. A232162.

r:=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
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)+r(n-2):
end if;
end proc:

LinearRecurrence[{1, 1, 3, 1}, {0, 1, 2, 5}, 31] (* Jean-François Alcover, Nov 26 2017 *)

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

a(n) = A232162(n) + A232162(n-1) + A232162(n-2).

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

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.

Original entry on oeis.org

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)

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User: Pamela E Harris

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.

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

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A232164 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 C and rank n.

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Formula

A232163 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra so(2n+1).

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Keywords

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Examples

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Crossrefs

Programs

Maple

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Formula

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula