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.

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

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 (0,0,0,1).

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

Cf. A319981, A319982, A320034, A320036.

Sequence in context: A304183 A157222 A367588 * A333527 A064047 A335152

Adjacent sequences: A320030 A320031 A320032 * A320034 A320035 A320036

Keyword

nonn

Author

Nick Mayers, Oct 03 2018

Status

approved