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A361044 Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1.

%I A361044 #11 Mar 18 2023 08:49:14
%S A361044 1,3,8,12,8,3,1,1,5,24,60,109,161,161,109,60,24,5,1,1,7,48,168,483,
%T A361044 1074,1805,2531,2886,2531,1805,1074,483,168,48,7,1
%N A361044 Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1.
%C A361044 The triangle is inspired by _Samuel J. Bevins_'s A360571.
%C A361044 The friendship graph is constructed by joining n copies of the cycle graph C_3 at a common vertex. F_1 is isomorphic to C_3 (the triangle graph) and has 3 vertices, F_2 is the butterfly graph and has 5 vertices and if n > 2 then F_n has 2*n + 1 vertices.
%H A361044 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 A361044 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 A361044 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>.
%H A361044 Wikipedia, <a href="https://en.wikipedia.org/wiki/Friendship_graph">Friendship Graph</a>.
%e A361044 The triangle T(n, k) starts:
%e A361044 [1] 1, 3, 8, 12, 8, 3, 1;
%e A361044 [2] 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1;
%e A361044 [3] 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1;
%o A361044 (SageMath)
%o A361044 from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras
%o A361044 def BettiNumbers(graph):
%o A361044     D = {}
%o A361044     for edge in graph.edges():
%o A361044         e = "x" + str(edge[0])
%o A361044         f = "x" + str(edge[1])
%o A361044         D[(e, f)] = {e + f : 1}
%o A361044     C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().
%o A361044          Stratified().Nilpotent())
%o A361044     L = LieAlgebra(QQ, D, nilpotent=True, category=C)
%o A361044     H = L.cohomology()
%o A361044     d = L.dimension() + 1
%o A361044     return [H[n].dimension() for n in range(d)]
%o A361044 def A361044_row(n):
%o A361044     return BettiNumbers(graphs.FriendshipGraph(n))
%o A361044 for n in range(1, 4): print(A361044_row(n))
%Y A361044 Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).
%K A361044 nonn,tabf,more
%O A361044 1,2
%A A361044 _Peter Luschny_, Mar 01 2023