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a(n) is the number of integer partitions of n for which the smallest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.
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%I #10 Sep 19 2018 06:17:47

%S 0,0,1,1,2,2,5,4,7,7,16,11,23,22,43,41,61,56,97,103,126,146,205,210,

%T 274,315,389,461,531,623,751,901,968,1227,1372,1661,1787,2238,2332,

%U 2998,3105,3921,4103,5241,5148,6778,6795,8745,8683,11231,11133,14523,14246,18284,18121,23536,22790,29627,29143,36990

%N a(n) is the number of integer partitions of n for which the smallest 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>2. To see this: for n odd, say n=2k+3, take the partition (2k+1,1,1); for n even, say n=2k+4, take the partition (2k+1,1,1,1).

%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, A318203

%K nonn

%O 1,5

%A _Nick Mayers_, Aug 20 2018