A318177 -id:A318177 - cp's OEIS frontend

A318176 a(n) is the number of integer partitions of n for which the greatest part minus the least part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

1, 1, 0, 0, 1, 1, 2, 3, 2, 4, 8, 4, 15, 12, 16, 21, 29, 30, 48, 40, 74, 67, 105, 102, 148, 154, 210, 223, 285, 292, 437, 428, 593, 630, 842, 894, 1168, 1317, 1628, 1759, 2249, 2426, 3112, 3356, 4158, 4637, 5647, 6172, 7657, 8400, 10146, 11401, 13450, 15069, 17948, 20108, 23674, 26867, 31398, 35133

Offset: 1

Nick Mayers, Aug 20 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)>0 for n=1,2 and n>4. To see this: for n=1,2 take the partitions (1) and (1,1), respectively; for n>3 odd take the partition (2,...,2,1,1,1); for n>2 congruent to 2 (mod 6), say n=6k+2, take the partition (2k+1,2k,2k,1); for n>4 congruent to 4 (mod 6), say n=6k+4, take the partition (2k+1,k+1,k+1,k+1,k); for n>0 congruent to 0 (mod 6), say n=6k, take the partition (2k,1,...,1) with 4k ones.

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A318177, A318178, A237832, A318196, A318203

A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 1, 2, 1, 8, 9, 5, 8, 15, 10, 17, 21, 24, 25, 45, 43, 68, 53, 82, 81, 143, 111, 165, 168, 247, 232, 314, 313, 442, 491, 587, 596, 918, 842, 1217, 1304, 1645, 1650, 2221, 2311, 2922, 3119, 4007, 4184, 5521, 5699, 7232, 7498, 9543, 9580, 12802

Offset: 1

Nick Mayers, Aug 20 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)>0 for n=2,6 and n>8. To see this: for n congruent to 2,6 (mod 8) take the partition of the form (2,...,2); for n>=9 congruent to 1,5 (mod 8), say n=4k+1, take the partition (4k-3,3,1); for n>7 congruent to 3 (mod 8), say n=8k+3, take the partition (4k,3,2,...,2) with 2k 2's; for n>7 congruent to 7 (mod 8) take the partition ((n-1)/2, (n-5)/2,3); for n>8 congruent to 4 (mod 8) take the partition (n-8,4,3,1); and for n>8 congruent to 0 (mod 8) take the partition (n-8,4,4).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A318176, A318177, A237832, A318196, A318203

A318196 a(n) is the number of integer partitions of n for which the smallest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 4, 7, 7, 16, 11, 23, 22, 43, 41, 61, 56, 97, 103, 126, 146, 205, 210, 274, 315, 389, 461, 531, 623, 751, 901, 968, 1227, 1372, 1661, 1787, 2238, 2332, 2998, 3105, 3921, 4103, 5241, 5148, 6778, 6795, 8745, 8683, 11231, 11133, 14523, 14246, 18284, 18121, 23536, 22790, 29627, 29143, 36990

Offset: 1

Nick Mayers, Aug 20 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)>0 for n>2. To see this: for n odd, say n=2k+3, take the partition (2k+1,1,1); for n even, say n=2k+4, take the partition (2k+1,1,1,1).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318203

A318203 a(n) is the number of integer partitions of n for which the largest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 3, 3, 3, 5, 6, 6, 10, 13, 18, 18, 27, 31, 40, 52, 58, 78, 95, 103, 136, 161, 194, 225, 265, 346, 386, 483, 585, 660, 797, 938, 1134, 1316, 1521, 1832, 2081, 2550, 2901, 3407, 3913, 4614, 5345, 6305, 7280, 8514, 9824, 11377, 13120, 14960, 17427, 19981, 23316, 26859, 30390

Offset: 1

Nick Mayers, Melissa Mayers Aug 21 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) > 0 for n = 3, 4 and n > 5. To see this: for n odd if n=3 take the partition (1,1,1), if n > 5 take the partition (2,...,2,1,1,1,1,1); for n > 2 congruent to 2 (mod 6), say n=6k+2, take the partition (2k,1,...,1) with 4k+2 1's; for n > 0 congruent to 4 (mod 6), say n=6k+4, take the partition (2k+1, 1,...,1) with 4k+3 1's; for n > 0 congruent to 0 (mod 6), say n=6k, take the partition (2k, 2k, 2k-1, 1).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318196

A318205 a(n) is the number of integer partitions of n for which the rank is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 5, 2, 7, 7, 6, 10, 12, 12, 16, 14, 22, 27, 28, 44, 52, 61, 76, 93, 112, 135, 162, 209, 243, 300, 350, 425, 484, 600, 662, 863, 964, 1153, 1351, 1629, 1874, 2244, 2584, 3074, 3507, 4213, 4805, 5725, 6524, 7742, 8770, 10357, 11813, 13936, 15704, 18445, 20896, 24552, 27724

Offset: 1

Nick Mayers, Aug 21 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)>0 for n>0. To see this for n, take the partition (n).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318196

A318238 a(n) is the number of integer partitions of n for which the crank is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 6, 5, 5, 8, 14, 15, 15, 24, 27, 38, 47, 58, 66, 83, 92, 118, 156, 187, 234, 262, 329, 367, 446, 517, 657, 712, 890, 1041, 1270, 1411, 1751, 1951, 2350, 2678, 3278, 3715

Offset: 1

Nick Mayers, Melissa Mayers, Aug 21 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([,]) 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)>0 for n>2. To see this: if n=k+1 take the partition (k,1).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318196, A318203

cp's OEIS Frontend

A318176 a(n) is the number of integer partitions of n for which the greatest part minus the least part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A318196 a(n) is the number of integer partitions of n for which the smallest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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Keywords

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A318203 a(n) is the number of integer partitions of n for which the largest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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Author

Keywords

Comments

Links

Crossrefs

A318205 a(n) is the number of integer partitions of n for which the rank is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Views

Author

Keywords

Comments

Links

Crossrefs

A318238 a(n) is the number of integer partitions of n for which the crank is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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Author

Keywords

Comments

Links

Crossrefs