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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..60.

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.

CROSSREFS

Cf. A318176, A318177, A237832, A318196, A318203

Sequence in context: A308831 A277327 A277328 * A283307 A273514 A048866

Adjacent sequences:  A318175 A318176 A318177 * A318179 A318180 A318181

KEYWORD

nonn

AUTHOR

Nick Mayers, Aug 20 2018

STATUS

approved

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Last modified December 10 20:38 EST 2019. Contains 329909 sequences. (Running on oeis4.)