cp's OEIS Frontend

A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

%I A318178 #14 Sep 19 2018 06:17:36
%S A318178 0,1,0,0,0,2,0,0,2,2,1,2,1,8,9,5,8,15,10,17,21,24,25,45,43,68,53,82,
%T A318178 81,143,111,165,168,247,232,314,313,442,491,587,596,918,842,1217,1304,
%U A318178 1645,1650,2221,2311,2922,3119,4007,4184,5521,5699,7232,7498,9543,9580,12802
%N A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.
%C A318178 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 A318178 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 A318178 a(n)>0 for n=2,6 and n>8. To see this: for n congruent to 2,6 (mod 8) take the partition of the form (2,...,2); for n>=9 congruent to 1,5 (mod 8), say n=4k+1, take the partition (4k-3,3,1); for n>7 congruent to 3 (mod 8), say n=8k+3, take the partition (4k,3,2,...,2) with 2k 2's; for n>7 congruent to 7 (mod 8) take the partition ((n-1)/2, (n-5)/2,3); for n>8 congruent to 4 (mod 8) take the partition (n-8,4,3,1); and for n>8 congruent to 0 (mod 8) take the partition (n-8,4,4).
%H A318178 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 A318178 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 A318178 Cf. A318176, A318177, A237832, A318196, A318203
%K A318178 nonn
%O A318178 1,6
%A A318178 _Nick Mayers_, Aug 20 2018