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%I #16 Sep 19 2018 06:09:38
%S 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,
%T 154,210,223,285,292,437,428,593,630,842,894,1168,1317,1628,1759,2249,
%U 2426,3112,3356,4158,4637,5647,6172,7657,8400,10146,11401,13450,15069,17948,20108,23674,26867,31398,35133
%N 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.
%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=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.
%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. A318177, A318178, A237832, A318196, A318203
%K nonn
%O 1,7
%A _Nick Mayers_, Aug 20 2018
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