A360625 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.

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, 440, 835, 1423, 1980, 1980, 1423, 835, 440, 176, 40, 5, 1, 1, 6, 70, 441, 1616, 4600, 11984, 26824, 46800, 63254, 70784, 70784, 63254, 46800, 26824, 11984, 4600, 1616, 441, 70, 6, 1

Offset: 1

Author

Samuel J. Bevins, Feb 14 2023

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  k= 0 1  2   3   4   5    6    7    8    9  10  11  12 13 14 15
n=1: 1 1
n=2: 1 2  2   1
n=3: 1 3  8  12   8   3    1
n=4: 1 4 20  56  84  90   84   56   20    4   1
n=5: 1 5 40 176 440 835 1423 1980 1980 1423 835 440 176 40  5  1
...
		

Links

• M. Aldi and S. Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
• M. Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
• Eric Weisstein's World of Mathematics, Complete Graph.

Crossrefs

Cf. A360571, A360572.

Programs

SageMath

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360625_row(n):
    if n == 1: return [1,1]
    return betti_numbers(LieAlgebraFromGraph(graphs.CompleteGraph(n)))