A319981 - OEIS

%I #33 Mar 04 2024 00:38:24

%S 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 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 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 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 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 2 for n>12.

%H Muniru A Asiru, <a href="/A319981/b319981.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_02">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1).

%F For n>12: a(n)=2 if n is odd, a(n)=0 if n is even.

%p 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 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 (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 Cf. A319982, A320033, A320034, A320036

%K nonn

%O 1,4

%A _Nick Mayers_, Oct 03 2018