A360936 -id:A360936 - cp's OEIS frontend

A360572 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the cycle graph on n vertices, n >= 3, 0 <= k <= 2*n.

1, 3, 8, 12, 8, 3, 1, 1, 4, 14, 25, 28, 25, 14, 4, 1, 1, 5, 20, 41, 70, 90, 70, 41, 20, 5, 1, 1, 6, 27, 68, 146, 219, 238, 219, 146, 68, 27, 6, 1, 1, 7, 35, 105, 259, 449, 644, 756, 644, 449, 259, 105, 35, 7, 1, 1, 8, 44, 152, 422, 857, 1476, 2012, 2172, 2012, 1476, 857, 422, 152, 44, 8, 1

Offset: 3

Samuel J. Bevins, Feb 12 2023

			Triangle begins:
   k=0 1  2   3   4   5    6    7    8    9   10  11  12  13  14 15 16
n=3  1 3  8  12   8   3    1
n=4  1 4 14  25  28  25   14    4    1
n=5  1 5 20  41  70  90   70   41   20    5    1
n=6  1 6 27  68 146 219  238  219  146   68   27   6   1
n=7  1 7 35 105 259 449  644  756  644  449  259 105  35   7   1
n=8  1 8 44 152 422 857 1476 2012 2172 2012 1476 857 422 152  44  8  1
  ...

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, Cycle Graph.

Cf. A360571 (path graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph)

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360936(n):
    return betti_numbers(LieAlgebraFromGraph(graphs.CycleGraph(n)))

A360937 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.

Original entry on oeis.org

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

Offset: 3

Samuel J. Bevins, Feb 26 2023

			Triangle T(n, k) begins:
   k=0 1  2   3  4     5    6    7    8    9   10   11  12  13 14 15 16
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 32 108 212  371  547  547  371  212  108   32   5   1
n=6: 1 6 45 171 442 1081 2025 2616 2722 2616 2025 1081 442 171 45  6  1
...

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, Wheel Graph.

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

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360937_row(n):
    return betti_numbers(LieAlgebraFromGraph(graphs.WheelGraph(n)))
for n in range(3, 7): print(A360937_row(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.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Mar 01 2023

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.

			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;

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.

Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).

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))

A361014 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the hypercube graph on 2^(n-1) vertices, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 4, 14, 25, 28, 25, 14, 4, 1, 1, 8, 64, 258, 986, 2870, 6134, 11586, 18830, 23832, 25078, 23832, 18830, 11586, 6134, 2870, 986, 258, 64, 8, 1

Offset: 1

Samuel J. Bevins, Feb 28 2023

			Triangle begins:
   k=0 1  2   3   4    5    6     7     8     9    10    11    12    13   14
n=1: 1 1
n=2: 1 2  2   1
n=3: 1 4 14  25  28   25   14     4     1
n=4: 1 8 64 258 986 2870 6134 11586 18830 23832 25078 23832 18830 11586 6134
...

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

Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360936_row(n):
    if n == 1: return [1, 1]
    return betti_numbers(LieAlgebraFromGraph(graphs.CubeGraph(n-1)))

cp's OEIS Frontend

A360572 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the cycle graph on n vertices, n >= 3, 0 <= k <= 2*n.

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A360937 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.

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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.

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SageMath

A361014 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the hypercube graph on 2^(n-1) vertices, n >= 1, k >= 0.

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Author

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SageMath