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A319982 a(n) is the number of integer partitions of n with largest part <= 4 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.

%I A319982 #26 Apr 21 2019 07:46:00
%S A319982 1,1,1,2,3,2,3,3,4,3,3,2,4,2,3,1,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,
%T A319982 3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,
%U A319982 4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0,4,2,3,0
%N A319982 a(n) is the number of integer partitions of n with largest part <= 4 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.
%C A319982 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 A319982 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 A319982 a(n) is periodic with period 4 for n > 16.
%H A319982 Colin Barker, <a href="/A319982/b319982.txt">Table of n, a(n) for n = 1..1000</a>
%H A319982 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 A319982 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 A319982 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A319982 For n > 16: a(n)=4 if 1 == n (mod 4), a(n)=2 if 2 == n (mod 4), a(n)=3 if 3 == n (mod 4), a(n)=0 if 0 == n (mod 4).
%F A319982 From _Colin Barker_, Apr 21 2019: (Start)
%F A319982 G.f.: x*(1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + x^8 + x^9 - x^11 - x^13 - x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)).
%F A319982 a(n) = a(n-4) for n>20.
%F A319982 (End)
%t A319982 Join[{1, 1, 1, 2, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 1}, LinearRecurrence[{0, 0, 0, 1}, {4, 2, 3, 0}, 100]] (* _Jean-François Alcover_, Dec 07 2018 *)
%o A319982 (PARI) Vec(x*(1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + x^8 + x^9 - x^11 - x^13 - x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^100)) \\ _Colin Barker_, Apr 21 2019
%Y A319982 Cf. A319981, A320033, A320034, A320036.
%K A319982 nonn
%O A319982 1,4
%A A319982 _Nick Mayers_, Oct 03 2018