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A320036 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.

%I A320036 #17 Mar 04 2024 10:37:52
%S A320036 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 A320036 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 A320036 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 A320036 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 A320036 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 A320036 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 A320036 a(n) is periodic with period 12 for n > 40.
%H A320036 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 A320036 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 A320036 <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 A320036 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 A320036 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 A320036 Cf. A319981, A319982, A320033, A320034.
%K A320036 nonn
%O A320036 1,4
%A A320036 _Nick Mayers_, Oct 03 2018
%E A320036 Missing term a(41)=7 inserted by _Jean-François Alcover_, Dec 08 2018