OFFSET
2,2
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().
EXAMPLE
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 ...
PROG
(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))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Samuel J. Bevins, Jan 11 2024
STATUS
approved