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A320034
a(n) is the number of integer partitions of n with largest part <= 6 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, 5, 6, 7, 7, 6, 9, 11, 8, 8, 9, 11, 9, 11, 9, 12, 9, 8, 9, 14, 6, 9, 5, 11, 6, 11, 4, 12, 5, 8, 4, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 3, 14, 5, 9, 3, 11, 5, 11, 3, 12, 5, 8, 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([_,_]), where [,] 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 12 for n > 36.
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 > 36: a(n)=14 if 1 == n (mod 12), a(n)=5 if 2,6,10 == n (mod 12), a(n)=9 if 3 == n (mod 12), a(n)=3 if 0,4,8 == n (mod 12), a(n)=11 if 5,7 == n (mod 12), a(n)=12 if 9 == n (mod 12), a(n)=8 if 11 == n (mod 12).
G.f.: x*(1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 11*x^11 + 9*x^12 + 8*x^13 + 7*x^14 + 4*x^15 + 6*x^16 + 2*x^17 + 5*x^18 + x^19 + 4*x^20 + x^21 + x^22 - 3*x^25 - 7*x^27 - 7*x^29 - 5*x^31 - 2*x^33 - 2*x^35 - x^37 - x^39 - x^41 - x^43) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)). - Colin Barker, Apr 21 2019
MATHEMATICA
a[n_] := If[n <= 36, {1, 1, 1, 2, 3, 3, 5, 6, 7, 7, 6, 9, 11, 8, 8, 9, 11, 9, 11, 9, 12, 9, 8, 9, 14, 6, 9, 5, 11, 6, 11, 4, 12, 5, 8, 4}[[n]], Switch[ Mod[n, 12], 1, 14, 2|6|10, 5, 3, 9, 0|4|8, 3, 5|7, 11, 9, 12, 11, 8]]; Array[a, 100] (* Jean-François Alcover, Dec 08 2018 *)
PROG
(PARI) Vec(x*(1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 11*x^11 + 9*x^12 + 8*x^13 + 7*x^14 + 4*x^15 + 6*x^16 + 2*x^17 + 5*x^18 + x^19 + 4*x^20 + x^21 + x^22 - 3*x^25 - 7*x^27 - 7*x^29 - 5*x^31 - 2*x^33 - 2*x^35 - x^37 - x^39 - x^41 - x^43) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)) + O(x^100)) \\ Colin Barker, Apr 21 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nick Mayers, Oct 03 2018
EXTENSIONS
Data corrected by Jean-François Alcover, Dec 08 2018
STATUS
approved