A319981 -id:A319981 - cp's OEIS frontend

A319982 a(n) is the number of integer partitions of n with largest part <= 4 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

1, 1, 1, 2, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 1, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0, 4, 2, 3, 0

Offset: 1

Nick Mayers, Oct 03 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n) is periodic with period 4 for n > 16.

Colin Barker, Table of n, a(n) for n = 1..1000
V. Coll, A. Mayers, N. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A319981, A320033, A320034, A320036.

Join[{1, 1, 1, 2, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 1}, LinearRecurrence[{0, 0, 0, 1}, {4, 2, 3, 0}, 100]] (* Jean-François Alcover, Dec 07 2018 *)

Vec(x*(1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + x^8 + x^9 - x^11 - x^13 - x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^100)) \\ Colin Barker, Apr 21 2019

For n > 16: a(n)=4 if 1 == n (mod 4), a(n)=2 if 2 == n (mod 4), a(n)=3 if 3 == n (mod 4), a(n)=0 if 0 == n (mod 4).
From Colin Barker, Apr 21 2019: (Start)
G.f.: x*(1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + x^8 + x^9 - x^11 - x^13 - x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)).
a(n) = a(n-4) for n>20.
(End)

A320033 a(n) is the number of integer partitions of n with largest part <= 5 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 5, 6, 6, 4, 6, 7, 5, 5, 6, 7, 4, 5, 5, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3

Offset: 1

Nick Mayers, Oct 03 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n) is periodic with period 4 for n > 20.

Colin Barker, Table of n, a(n) for n = 1..1000
V. Coll, A. Mayers, N. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A319981, A319982, A320034, A320036.

Join[{1, 1, 1, 2, 3, 3, 4, 5, 6, 6, 4, 6, 7, 5, 5, 6, 7, 4, 5, 5}, LinearRecurrence[{0, 0, 0, 1}, {7, 3, 5, 3}, 100]] (* Jean-François Alcover, Dec 07 2018 *)

Vec(x*(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + x^11 + x^12 - x^13 + x^14 - x^17 - x^19 - x^21 - 2*x^23) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^100)) \\ Colin Barker, Apr 21 2019

For n > 20: a(n)=7 if 1 == n (mod 4), a(n)=3 if 2 == n (mod 4), a(n)=5 if 3 == n (mod 4), a(n)=3 if 0 == n (mod 4).
From Colin Barker, Apr 21 2019: (Start)
G.f.: x*(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + x^11 + x^12 - x^13 + x^14 - x^17 - x^19 - x^21 - 2*x^23) / ((1 - x)*(1 + x)*(1 + x^2)).
a(n) = a(n-4) for n>24.
(End)

A320034 a(n) is the number of integer partitions of n with largest part <= 6 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 7, 7, 6, 9, 11, 8, 8, 9, 11, 9, 11, 9, 12, 9, 8, 9, 14, 6, 9, 5, 11, 6, 11, 4, 12, 5, 8, 4, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3

Offset: 1

Nick Mayers, Oct 03 2018

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]), where [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n) is periodic with period 12 for n > 36.

Colin Barker, Table of n, a(n) for n = 1..1000
V. Coll, A. Mayers, N. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,0,0,1,0,1).

Cf. A319981, A319982, A320033, A320036.

a[n_] := If[n <= 36, {1, 1, 1, 2, 3, 3, 5, 6, 7, 7, 6, 9, 11, 8, 8, 9, 11, 9, 11, 9, 12, 9, 8, 9, 14, 6, 9, 5, 11, 6, 11, 4, 12, 5, 8, 4}[[n]], Switch[ Mod[n, 12], 1, 14, 2|6|10, 5, 3, 9, 0|4|8, 3, 5|7, 11, 9, 12, 11, 8]]; Array[a, 100] (* Jean-François Alcover, Dec 08 2018 *)

Vec(x*(1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 11*x^11 + 9*x^12 + 8*x^13 + 7*x^14 + 4*x^15 + 6*x^16 + 2*x^17 + 5*x^18 + x^19 + 4*x^20 + x^21 + x^22 - 3*x^25 - 7*x^27 - 7*x^29 - 5*x^31 - 2*x^33 - 2*x^35 - x^37 - x^39 - x^41 - x^43) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)) + O(x^100)) \\ Colin Barker, Apr 21 2019

For n > 36: a(n)=14 if 1 == n (mod 12), a(n)=5 if 2,6,10 == n (mod 12), a(n)=9 if 3 == n (mod 12), a(n)=3 if 0,4,8 == n (mod 12), a(n)=11 if 5,7 == n (mod 12), a(n)=12 if 9 == n (mod 12), a(n)=8 if 11 == n (mod 12).

G.f.: x*(1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 11*x^11 + 9*x^12 + 8*x^13 + 7*x^14 + 4*x^15 + 6*x^16 + 2*x^17 + 5*x^18 + x^19 + 4*x^20 + x^21 + x^22 - 3*x^25 - 7*x^27 - 7*x^29 - 5*x^31 - 2*x^33 - 2*x^35 - x^37 - x^39 - x^41 - x^43) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)). - Colin Barker, Apr 21 2019

Data corrected by Jean-François Alcover, Dec 08 2018

A320036 a(n) is the number of integer partitions of n with largest part <= 7 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 8, 8, 8, 12, 14, 13, 11, 16, 16, 16, 17, 18, 18, 20, 16, 21, 20, 19, 16, 17, 19, 15, 18, 15, 19, 12, 17, 14, 20, 11, 17, 10, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7, 19, 9, 18, 7, 19, 9, 17, 7, 20, 9, 17, 7

Offset: 1

Nick Mayers, Oct 03 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n) is periodic with period 12 for n > 40.

V. Coll, A. Mayers, N. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
Index entries for linear recurrences with constant coefficients, signature (0, -1, 0, 0, 0, 1, 0, 1).

Cf. A319981, A319982, A320033, A320034.

a[n_] := If[n <= 40, {1, 1, 1, 2, 3, 3, 5, 7, 8, 8, 8, 12, 14, 13, 11, 16, 16, 16, 17, 18, 18, 20, 16, 21, 20, 19, 16, 17, 19, 15, 18, 15, 19, 12, 17, 14, 20, 11, 17, 10}[[n]], Switch[Mod[n, 12], 1|5|9, 7, 2, 20, 3|7|11, 9, 0|4, 17, 6|10, 19, 8, 18]]; Array[a, 125] (* Jean-François Alcover, Dec 08 2018 *)

For n > 40: a(n)=7 if 1,5,9 == n (mod 12), a(n)=20 if 2 == n (mod 12), a(n)=9 if 3,7,11 == n (mod 12), a(n)=17 if 0,4 == n (mod 12), a(n)=19 if 6,10 == n (mod 12), a(n)=18 if 8 == n (mod 12).

Missing term a(41)=7 inserted by Jean-François Alcover, Dec 08 2018

cp's OEIS Frontend

A319982 a(n) is the number of integer partitions of n with largest part <= 4 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

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A320033 a(n) is the number of integer partitions of n with largest part <= 5 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

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A320034 a(n) is the number of integer partitions of n with largest part <= 6 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

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A320036 a(n) is the number of integer partitions of n with largest part <= 7 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

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