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%I #26 Dec 18 2023 10:09:43
%S 0,0,0,1,0,1,1,2,3,2,2,5,5,8,8,11,18,20,26,26,35,49,56,73,88,101,130,
%T 148,182,207,260,310,385,455,579,657,800,910,1135,1310,1546,1763,2169,
%U 2488,2936,3352,3962,4612,5435,6187,7370,8430,9951,11276,13236,15133,17624,20009,23551,26464
%N a(n) is the number of integer partitions of n for which the Kimberling index 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=4 and n>5. To see this: for n>0 congruent to 0 (mod 4), say 4k+4, take the partition of the form (2k+3,2k+1); for n congruent to 2 (mod 4) if n=6 take (4,4,1), if n=10 take (5,3,2), if n>10, say n=4k+10, take the partition (2k+7,2k-1,1,1,1,1); for n>1 congruent to 1 (mod 6), say n=6k+1, take the partition (2k+3,2k-1,2k-1); for n>5 congruent to 5 (mod 6), say n=6k+5, take the partition (2k+3,2k+3,2k-1); for n>3 congruent to 3 (mod 6), say n=6k-3, take the partition (2k+1,2,...,2) with 2k-2 2's.
%H George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%H V. Coll, M. Hyatt, C. Magnant, and 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 and 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. A318176, A318178, A237832, A318196, A318203
%K nonn
%O 1,8
%A _Nick Mayers_ and _Melissa Mayers_, Aug 20 2018
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