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.

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

Offset: 1

Author

Nick Mayers and Melissa Mayers, Aug 20 2018

Keywords

• nonn

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

• George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
• V. Coll, M. Hyatt, C. Magnant, and H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
• V. Dergachev and A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Crossrefs

Cf. A318176, A318178, A237832, A318196, A318203