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%I #7 Sep 19 2018 06:18:10
%S 1,1,2,1,1,2,2,5,2,7,7,6,10,12,12,16,14,22,27,28,44,52,61,76,93,112,
%T 135,162,209,243,300,350,425,484,600,662,863,964,1153,1351,1629,1874,
%U 2244,2584,3074,3507,4213,4805,5725,6524,7742,8770,10357,11813,13936,15704,18445,20896,24552,27724
%N a(n) is the number of integer partitions of n for which the rank 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>0. To see this for n, take the partition (n).
%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 1,3
%A _Nick Mayers_, Aug 21 2018
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