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A318176 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.

%I A318176 #16 Sep 19 2018 06:09:38
%S A318176 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 A318176 154,210,223,285,292,437,428,593,630,842,894,1168,1317,1628,1759,2249,
%U A318176 2426,3112,3356,4158,4637,5647,6172,7657,8400,10146,11401,13450,15069,17948,20108,23674,26867,31398,35133
%N A318176 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 A318176 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 A318176 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 A318176 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 A318176 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 A318176 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 A318176 Cf. A318177, A318178, A237832, A318196, A318203
%K A318176 nonn
%O A318176 1,7
%A A318176 _Nick Mayers_, Aug 20 2018