%I #24 Feb 06 2024 11:55:55

%S 1,2,2,1,1,4,11,16,16,11,4,1,1,7,33,95,212,344,444,444,344,212,95,33,

%T 7,1,1,12,90,454,1780,5489,14036,29804,54007,83404,111361,128378,

%U 128378,111361,83404,54007,29804,14036,5489,1780,454,90,12,1

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

%H Marco Aldi and Samuel Bevins, <a href="https://arxiv.org/abs/2212.13608">2-step Nilpotent L_oo-algebras and Hypergraphs</a>, arXiv:2212.13608 [math.CO], 2023. See page 9.

%H Meera Mainkar, <a href="https://arxiv.org/abs/1310.3414">Graphs and two step nilpotent Lie algebras</a>, arXiv:1310.3414 [math.DG], 2013. See page 1.

%H SageMath Graph Theory, <a href="https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/generators/families.html">Various families of graphs</a>, see FibonacciTree().

%e Triangle begins:

%e k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

%e n=2: 1 2 2 1

%e n=3: 1 4 11 16 16 11 4 1

%e n=4: 1 7 33 95 212 344 444 444 344 212 95 33 7 1

%e n=5: 1 12 90 454 1780 5489 14036 29804 54007 83404 111361 128378 128378 111361 83404 54007 ...

%o (SageMath)

%o from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras

%o def BettiNumbers(graph):

%o D = {}

%o for edge in graph.edges():

%o e = "x" + str(edge[0])

%o f = "x" + str(edge[1])

%o D[(e, f)] = {e + f : 1}

%o C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().

%o Stratified().Nilpotent())

%o L = LieAlgebra(QQ, D, nilpotent=True, category=C)

%o H = L.cohomology()

%o d = L.dimension() + 1

%o return [H[n].dimension() for n in range(d)]

%o # Example usage:

%o n = 5

%o X = BettiNumbers(graphs.FibonacciTree(n))

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

%K nonn,tabf

%O 2,2

%A _Samuel J. Bevins_, Jan 11 2024