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.

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

Colin Barker, Table of n, a(n) for n = 1..1000

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.

Index entries for linear recurrences with constant coefficients, signature (0,-1,0,0,0,1,0,1).

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

Cf. A319981, A319982, A320033, A320036.

Sequence in context: A255563 A331288 A115350 * A262882 A187043 A081211

Adjacent sequences: A320031 A320032 A320033 * A320035 A320036 A320037

Keyword

nonn,easy

Author

Nick Mayers, Oct 03 2018

Extensions

Data corrected by Jean-François Alcover, Dec 08 2018

Status

approved