A368135 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the Fibonacci trees of order n >= 2.

1, 2, 2, 1, 1, 4, 11, 16, 16, 11, 4, 1, 1, 7, 33, 95, 212, 344, 444, 444, 344, 212, 95, 33, 7, 1, 1, 12, 90, 454, 1780, 5489, 14036, 29804, 54007, 83404, 111361, 128378, 128378, 111361, 83404, 54007, 29804, 14036, 5489, 1780, 454, 90, 12, 1

Offset: 2

Author

Samuel J. Bevins, Jan 11 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  k=0 1  2  3   4   5    6    7    8    9    10    11   12    13   14   15
n=2: 1 2   2  1
n=3: 1 4  11  16   16   11     4     1
n=4: 1 7  33  95  212  344   444   444   344   212     95     33      7      1
n=5: 1 12 90 454 1780 5489 14036 29804 54007 83404 111361 128378 128378 111361 83404 54007 ...
		

Links

• Marco Aldi and Samuel Bevins, 2-step Nilpotent L_oo-algebras and Hypergraphs, arXiv:2212.13608 [math.CO], 2023. See page 9.
• Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
• SageMath Graph Theory, Various families of graphs, see FibonacciTree().

Crossrefs

Cf. A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360938 (ladder graph), A360937 (wheel graph).

Programs

SageMath

from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras
def BettiNumbers(graph):
    D = {}
    for edge in graph.edges():
        e = "x" + str(edge[0])
        f = "x" + str(edge[1])
        D[(e, f)] = {e + f : 1}
    C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().
         Stratified().Nilpotent())
    L = LieAlgebra(QQ, D, nilpotent=True, category=C)
    H = L.cohomology()
    d = L.dimension() + 1
    return [H[n].dimension() for n in range(d)]
# Example usage:
n = 5
X = BettiNumbers(graphs.FibonacciTree(n))