A318178 - OEIS

%I #14 Sep 19 2018 06:17:36

%S 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,

%T 81,143,111,165,168,247,232,314,313,442,491,587,596,918,842,1217,1304,

%U 1645,1650,2221,2311,2922,3119,4007,4184,5521,5699,7232,7498,9543,9580,12802

%N 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.

%C 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.

%C 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.

%C 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).

%H V. Coll, M. Hyatt, C. Magnant, H. Wang, <a href="http://dx.doi.org/10.4172/1736-4337.1000227">Meander graphs and Frobenius seaweed Lie algebras II</a>, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.

%H V. Dergachev, A. Kirillov, <a href="https://www.emis.de/journals/JLT/vol.10_no.2/6.html">Index of Lie algebras of seaweed type</a>, J. Lie Theory 10 (2) (2000) 331-343.

%Y Cf. A318176, A318177, A237832, A318196, A318203

%K nonn

%O 1,6

%A _Nick Mayers_, Aug 20 2018