# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a368135 Showing 1-1 of 1 %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, 2-step Nilpotent L_oo-algebras and Hypergraphs, arXiv:2212.13608 [math.CO], 2023. See page 9. %H A368135 Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1. %H A368135 SageMath Graph Theory, Various families of graphs, 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE