

A361044


Triangle read by rows. T(n, k) is the kth LieBetti 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
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OFFSET

1,2


COMMENTS

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



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)]
return BettiNumbers(graphs.FriendshipGraph(n))
for n in range(1, 4): print(A361044_row(n))


CROSSREFS



KEYWORD

nonn,tabf,more


AUTHOR



STATUS

approved



