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%I #17 Mar 04 2024 10:37:52
%S 1,1,1,2,3,3,5,7,8,8,8,12,14,13,11,16,16,16,17,18,18,20,16,21,20,19,
%T 16,17,19,15,18,15,19,12,17,14,20,11,17,10,7,19,9,18,7,19,9,17,7,20,9,
%U 17,7,19,9,18,7,19,9,17,7,20,9,17,7,19,9,18,7,19,9,17,7,20,9,17,7,19,9,18,7,19,9,17,7,20,9,17,7,19,9,18,7,19,9,17,7,20,9,17,7,19,9,18,7,19,9,17,7,20,9,17,7,19,9,18,7,19,9,17,7,20,9,17,7
%N a(n) is the number of integer partitions of n with largest part <= 7 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 12 for n > 40.
%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_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, -1, 0, 0, 0, 1, 0, 1).
%F For n > 40: a(n)=7 if 1,5,9 == n (mod 12), a(n)=20 if 2 == n (mod 12), a(n)=9 if 3,7,11 == n (mod 12), a(n)=17 if 0,4 == n (mod 12), a(n)=19 if 6,10 == n (mod 12), a(n)=18 if 8 == n (mod 12).
%t a[n_] := If[n <= 40, {1, 1, 1, 2, 3, 3, 5, 7, 8, 8, 8, 12, 14, 13, 11, 16, 16, 16, 17, 18, 18, 20, 16, 21, 20, 19, 16, 17, 19, 15, 18, 15, 19, 12, 17, 14, 20, 11, 17, 10}[[n]], Switch[Mod[n, 12], 1|5|9, 7, 2, 20, 3|7|11, 9, 0|4, 17, 6|10, 19, 8, 18]]; Array[a, 125] (* _Jean-François Alcover_, Dec 08 2018 *)
%Y Cf. A319981, A319982, A320033, A320034.
%K nonn
%O 1,4
%A _Nick Mayers_, Oct 03 2018
%E Missing term a(41)=7 inserted by _Jean-François Alcover_, Dec 08 2018
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