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A368135
Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the Fibonacci trees of order n >= 2.
0
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
2,2
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
Cf. A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360938 (ladder graph), A360937 (wheel graph).
Sequence in context: A176663 A113021 A298261 * A341991 A152937 A331315
KEYWORD
nonn,tabf
AUTHOR
Samuel J. Bevins, Jan 11 2024
STATUS
approved