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A318177 a(n) is the number of integer partitions of n for which the Kimberling index is equal to the index of the seaweed algebra formed by the integer partition paired with its weight. 6
0, 0, 0, 1, 0, 1, 1, 2, 3, 2, 2, 5, 5, 8, 8, 11, 18, 20, 26, 26, 35, 49, 56, 73, 88, 101, 130, 148, 182, 207, 260, 310, 385, 455, 579, 657, 800, 910, 1135, 1310, 1546, 1763, 2169, 2488, 2936, 3352, 3962, 4612, 5435, 6187, 7370, 8430, 9951, 11276, 13236, 15133, 17624, 20009, 23551, 26464 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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=4 and n>5. To see this: for n>0 congruent to 0 (mod 4), say 4k+4, take the partition of the form (2k+3,2k+1); for n congruent to 2 (mod 4) if n=6 take (4,4,1), if n=10 take (5,3,2), if n>10, say n=4k+10, take the partition (2k+7,2k-1,1,1,1,1); for n>1 congruent to 1 (mod 6), say n=6k+1, take the partition (2k+3,2k-1,2k-1); for n>5 congruent to 5 (mod 6), say n=6k+5, take the partition (2k+3,2k+3,2k-1); for n>3 congruent to 3 (mod 6), say n=6k-3, take the partition (2k+1,2,...,2) with 2k-2 2's.

LINKS

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

G. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint.

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, A318178, A237832, A318196, A318203

Sequence in context: A238689 A166454 A283239 * A128651 A093797 A214320

Adjacent sequences:  A318174 A318175 A318176 * A318178 A318179 A318180

KEYWORD

nonn

AUTHOR

Nick Mayers and Melissa Mayers, Aug 20 2018

STATUS

approved

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)