The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A361044 Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1. 1
 1, 3, 8, 12, 8, 3, 1, 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1, 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The triangle is inspired by Samuel J. Bevins's A360571. 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. LINKS Table of n, a(n) for n=1..36. Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9. Meera G. Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1. Eric Weisstein's World of Mathematics, Dutch Windmill Graph. Wikipedia, Friendship Graph. EXAMPLE The triangle T(n, k) starts: [1] 1, 3, 8, 12, 8, 3, 1; [2] 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1; [3] 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1; PROG (SageMath) from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras def BettiNumbers(graph): D = {} for edge in graph.edges(): e = "x" + str(edge[0]) f = "x" + str(edge[1]) D[(e, f)] = {e + f : 1} C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional(). Stratified().Nilpotent()) L = LieAlgebra(QQ, D, nilpotent=True, category=C) H = L.cohomology() d = L.dimension() + 1 return [H[n].dimension() for n in range(d)] def A361044_row(n): return BettiNumbers(graphs.FriendshipGraph(n)) for n in range(1, 4): print(A361044_row(n)) CROSSREFS Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph). Sequence in context: A050391 A360572 A360937 * A288865 A331069 A161517 Adjacent sequences: A361041 A361042 A361043 * A361045 A361046 A361047 KEYWORD nonn,tabf,more AUTHOR Peter Luschny, Mar 01 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 21:37 EST 2023. Contains 367662 sequences. (Running on oeis4.)