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%I #10 Sep 19 2018 06:17:59
%S 0,0,1,1,0,1,1,3,3,3,5,6,6,10,13,18,18,27,31,40,52,58,78,95,103,136,
%T 161,194,225,265,346,386,483,585,660,797,938,1134,1316,1521,1832,2081,
%U 2550,2901,3407,3913,4614,5345,6305,7280,8514,9824,11377,13120,14960,17427,19981,23316,26859,30390
%N a(n) is the number of integer partitions of n for which the largest part 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 = 3, 4 and n > 5. To see this: for n odd if n=3 take the partition (1,1,1), if n > 5 take the partition (2,...,2,1,1,1,1,1); for n > 2 congruent to 2 (mod 6), say n=6k+2, take the partition (2k,1,...,1) with 4k+2 1's; for n > 0 congruent to 4 (mod 6), say n=6k+4, take the partition (2k+1, 1,...,1) with 4k+3 1's; for n > 0 congruent to 0 (mod 6), say n=6k, take the partition (2k, 2k, 2k-1, 1).
%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. A237832, A318176, A318177, A318178, A318196
%K nonn
%O 1,8
%A _Nick Mayers_, _Melissa Mayers_ Aug 21 2018
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