|
%I #28 Feb 28 2023 16:35:53
%S 1,1,1,2,2,1,1,3,8,12,8,3,1,1,4,20,56,84,90,84,56,20,4,1,1,5,40,176,
%T 440,835,1423,1980,1980,1423,835,440,176,40,5,1,1,6,70,441,1616,4600,
%U 11984,26824,46800,63254,70784,70784,63254,46800,26824,11984,4600,1616,441,70,6,1
%N Triangle read by rows: T(n,k) is the k-th Lie-Betti number of a complete graph on n vertices, n >= 1, k >= 0.
%H M. Aldi and S. Bevins, <a href="https://arxiv.org/abs/2212.13608">L_oo-algebras and hypergraphs</a>, arXiv:2212.13608 [math.CO], 2022. See page 9.
%H M. 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.
%e Triangle begins:
%e k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e n=1: 1 1
%e n=2: 1 2 2 1
%e n=3: 1 3 8 12 8 3 1
%e n=4: 1 4 20 56 84 90 84 56 20 4 1
%e n=5: 1 5 40 176 440 835 1423 1980 1980 1423 835 440 176 40 5 1
%e ...
%o (SageMath) # uses[betti_numbers, LieAlgebraFromGraph from A360571]
%o def A360625_row(n):
%o if n == 1: return [1,1]
%o return betti_numbers(LieAlgebraFromGraph(graphs.CompleteGraph(n)))
%Y Cf. A360571, A360572.
%K nonn,tabf
%O 1,4
%A _Samuel J. Bevins_, Feb 14 2023
|
