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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.
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%I #26 Apr 21 2019 07:46:00

%S 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 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 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 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 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) is periodic with period 4 for n > 16.

%H Colin Barker, <a href="/A319982/b319982.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

%F 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 From _Colin Barker_, Apr 21 2019: (Start)

%F 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 a(n) = a(n-4) for n>20.

%F (End)

%t 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 (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 Cf. A319981, A320033, A320034, A320036.

%K nonn

%O 1,4

%A _Nick Mayers_, Oct 03 2018