A360571 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the path graph on n-vertices, n >= 1, 0 <= k <= 2*n - 1.
1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 11, 16, 16, 11, 4, 1, 1, 5, 17, 33, 48, 48, 33, 17, 5, 1, 1, 6, 24, 58, 107, 140, 140, 107, 58, 24, 6, 1, 1, 7, 32, 92, 203, 329, 424, 424, 329, 203, 92, 32, 7, 1, 1, 8, 41, 136, 347, 668, 1039, 1280, 1280, 1039, 668, 347, 136, 41, 8, 1
Offset: 1
Examples
Triangle begins:
k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n=1: 1 1
n=2: 1 2 2 1
n=3: 1 3 6 6 3 1
n=4: 1 4 11 16 16 11 4 1
n=5: 1 5 17 33 48 48 33 17 5 1
n=6: 1 6 24 58 107 140 140 107 58 24 6 1
n=7: 1 7 32 92 203 329 424 424 329 203 92 32 7 1
n=8: 1 8 41 136 347 668 1039 1280 1280 1039 668 347 136 41 8 1
Links
- 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, Path Graph.
Crossrefs
Programs
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SageMath
from sage.algebras.lie_algebras.lie_algebra import LieAlgebra def LieAlgebraFromGraph(G, Module = QQ): ''' Takes a graph and a module (optional) as an input.''' d = {} for edge in G.edges(): # this defines the relations among the generators of the Lie algebra key = ("x" + str(edge[0]), "x" + str(edge[1])) #[x_i, x_j] value = {"x_" + str(edge[0]) + "" + str(edge[1]): 1} #x{i,j} d[key] = value #appending to the dictionary d C = LieAlgebras(Module).WithBasis().Graded() #defines the category that we need to work with. C = C.FiniteDimensional().Stratified().Nilpotent() #specifies that the algebras we want should be finite, stratified, and nilpotent L = LieAlgebra(Module, d, nilpotent=True, category=C) def sort_generators_by_grading(lie_algebra, grading_operator): #this sorts the generators by their grading. In this case, V1 are vertices and V2 generators = lie_algebra.gens() grading = [grading_operator(g) for g in generators] #using the grading operator to split the elements into their respective vector spaces sorted_generators = [g for _, g in sorted(zip(grading, generators))] grouped_generators = {} for g in sorted_generators: if grading_operator(g) in grouped_generators: grouped_generators[grading_operator(g)].append(g) else: grouped_generators[grading_operator(g)] = [g] return grouped_generators grading_operator = lambda g: g.degree() #defining the grading operator grouped_generators = sort_generators_by_grading(L, grading_operator) #evaluating the function to pull the generators apart V1 = grouped_generators[1] #elements from vertices V2 = grouped_generators[2] #elements from edges return L #, V1, V2 #returns the Lie algebra and the two vector spaces def betti_numbers(lie_algebra): #this function will calculate the Lie theoretic Betti numbers and return them as a list dims = [] H = lie_algebra.cohomology() for n in range(lie_algebra.dimension() + 1): dims.append(H[n].dimension()) return dims def A360571_row(n): if n == 1: return [1, 1] return betti_numbers(LieAlgebraFromGraph(graphs.PathGraph(n))) for n in range(1,7): print(A360571_row(n))