A360937 - OEIS

%I #19 May 04 2024 14:55:54

%S 1,3,8,12,8,3,1,1,4,20,56,84,90,84,56,20,4,1,1,5,32,108,212,371,547,

%T 547,371,212,108,32,5,1,1,6,45,171,442,1081,2025,2616,2722,2616,2025,

%U 1081,442,171,45,6,1,1,7,60,258,842,2489,5440,8855,12955,16785,16785,12955,8855,5440,2489,842,258,60,7,1

%N Triangle read by rows: T(n, k) is the k-th Lie-Betti number of a wheel graph on n vertices, for n >= 3 and k >= 0.

%H Marco Aldi and Samuel 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 Meera G. 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="https://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>.

%e Triangle T(n, k) begins:

%e    k=0 1  2   3  4     5    6    7    8    9   10   11  12  13 14 15 16

%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 32 108 212  371  547  547  371  212  108   32   5   1

%e n=6: 1 6 45 171 442 1081 2025 2616 2722 2616 2025 1081 442 171 45  6  1

%e ...

%o (SageMath) # uses[betti_numbers, LieAlgebraFromGraph from A360571]

%o def A360937_row(n):

%o     return betti_numbers(LieAlgebraFromGraph(graphs.WheelGraph(n)))

%o for n in range(3, 7): print(A360937_row(n))

%Y Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A361044 (friendship graph).

%K nonn,tabf

%O 3,2

%A _Samuel J. Bevins_, Feb 26 2023