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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.

%I A368135 #24 Feb 06 2024 11:55:55
%S A368135 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 A368135 7,1,1,12,90,454,1780,5489,14036,29804,54007,83404,111361,128378,
%U A368135 128378,111361,83404,54007,29804,14036,5489,1780,454,90,12,1
%N A368135 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the Fibonacci trees of order n >= 2.
%H A368135 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 A368135 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 A368135 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 A368135 Triangle begins:
%e A368135   k=0 1  2  3   4   5    6    7    8    9    10    11   12    13   14   15
%e A368135 n=2: 1 2   2  1
%e A368135 n=3: 1 4  11  16   16   11     4     1
%e A368135 n=4: 1 7  33  95  212  344   444   444   344   212     95     33      7      1
%e A368135 n=5: 1 12 90 454 1780 5489 14036 29804 54007 83404 111361 128378 128378 111361 83404 54007 ...
%o A368135 (SageMath)
%o A368135 from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras
%o A368135 def BettiNumbers(graph):
%o A368135     D = {}
%o A368135     for edge in graph.edges():
%o A368135         e = "x" + str(edge[0])
%o A368135         f = "x" + str(edge[1])
%o A368135         D[(e, f)] = {e + f : 1}
%o A368135     C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().
%o A368135          Stratified().Nilpotent())
%o A368135     L = LieAlgebra(QQ, D, nilpotent=True, category=C)
%o A368135     H = L.cohomology()
%o A368135     d = L.dimension() + 1
%o A368135     return [H[n].dimension() for n in range(d)]
%o A368135 # Example usage:
%o A368135 n = 5
%o A368135 X = BettiNumbers(graphs.FibonacciTree(n))
%Y A368135 Cf. A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360938 (ladder graph), A360937 (wheel graph).
%K A368135 nonn,tabf
%O A368135 2,2
%A A368135 _Samuel J. Bevins_, Jan 11 2024