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

1,7

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=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.

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. A318177, A318178, A237832, A318196, A318203

Sequence in context: A274486 A227961 A108838 * A105070 A154578 A059576

Adjacent sequences: A318173 A318174 A318175 * A318177 A318178 A318179

Keyword

nonn

Author

Nick Mayers, Aug 20 2018

Status

approved