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URL: https://oeis.org/A320033

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A320033
a(n) is the number of integer partitions of n with largest part <= 5 for which the index of the seaweed algebra formed by the integer partition paired with its weight is 0.
5
1, 1, 1, 2, 3, 3, 4, 5, 6, 6, 4, 6, 7, 5, 5, 6, 7, 4, 5, 5, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3
OFFSET
1,4
COMMENTS
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.
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.
a(n) is periodic with period 4 for n > 20.
LINKS
V. Coll, A. Mayers, N. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv preprint arXiv:1809.09271 [math.CO], 2018.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
FORMULA
For n > 20: a(n)=7 if 1 == n (mod 4), a(n)=3 if 2 == n (mod 4), a(n)=5 if 3 == n (mod 4), a(n)=3 if 0 == n (mod 4).
From Colin Barker, Apr 21 2019: (Start)
G.f.: x*(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + x^11 + x^12 - x^13 + x^14 - x^17 - x^19 - x^21 - 2*x^23) / ((1 - x)*(1 + x)*(1 + x^2)).
a(n) = a(n-4) for n>24.
(End)
MATHEMATICA
Join[{1, 1, 1, 2, 3, 3, 4, 5, 6, 6, 4, 6, 7, 5, 5, 6, 7, 4, 5, 5}, LinearRecurrence[{0, 0, 0, 1}, {7, 3, 5, 3}, 100]] (* Jean-François Alcover, Dec 07 2018 *)
PROG
(PARI) Vec(x*(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + x^11 + x^12 - x^13 + x^14 - x^17 - x^19 - x^21 - 2*x^23) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^100)) \\ Colin Barker, Apr 21 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Nick Mayers, Oct 03 2018
STATUS
approved