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A319981 a(n) is the number of integer partitions of n with largest part <= 3 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

%I A319981 #33 Mar 04 2024 00:38:24
%S A319981 1,1,1,2,2,1,2,1,2,1,2,1,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,
%T A319981 2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,
%U A319981 2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2
%N A319981 a(n) is the number of integer partitions of n with largest part <= 3 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.
%C A319981 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 A319981 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 A319981 a(n) is periodic with period 2 for n>12.
%H A319981 Muniru A Asiru, <a href="/A319981/b319981.txt">Table of n, a(n) for n = 1..1000</a>
%H A319981 V. Coll, A. Mayers, N. Mayers, <a href="https://arxiv.org/abs/1809.09271">Statistics on integer partitions arising from seaweed algebras</a>, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
%H A319981 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.
%H A319981 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1).
%F A319981 For n>12: a(n)=2 if n is odd, a(n)=0 if n is even.
%p A319981 1,1,1,2,2,1,2,1,2,1,2,1,2 seq(op([0,2]),n=1..30); # _Muniru A Asiru_, Dec 07 2018
%t A319981 Join[{1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1}, LinearRecurrence[{0, 1}, {2, 0}, 100]] (* _Jean-François Alcover_, Dec 07 2018 *)
%o A319981 (GAP) a:=[1,1,1,2,2,1,2,1,2,1,2,1,2];; Concatenation(a,Flat(List([1..30],n->[0,2]))); # _Muniru A Asiru_, Dec 07 2018
%Y A319981 Cf. A319982, A320033, A320034, A320036
%K A319981 nonn
%O A319981 1,4
%A A319981 _Nick Mayers_, Oct 03 2018