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%I #23 Dec 03 2021 17:06:46
%S 0,0,1,1,2,11,26,122,781
%N a(n) is the number of connected Veblen hypergraphs (i.e., k-uniform hypergraphs where the degree of each vertex is divisible by k) with n edges up to isomorphism.
%H Gregory Clark and Joshua Cooper, <a href="https://arxiv.org/abs/1812.00468">A Harary-Sachs Theorem for Hypergraphs</a>, arXiv:1812.00468 [math.CO], 2018.
%H Gregory J. Clark and Joshua Cooper, <a href="https://arxiv.org/abs/2107.10781">Applications of the Harary-Sachs Theorem for Hypergraphs</a>, arXiv:2107.10781 [math.CO], 2021.
%H J. Cooper and A. Dutle, <a href="https://doi.org/10.1016/j.laa.2011.11.018">Spectra of uniform hypergraphs</a>, Linear Algebra Appl. 436 (2012) 3268-3292.
%e The only 3-uniform Veblen hypergraph with 3 edges is the single edge with multiplicity 3, {(1,2,3)^3}.
%e The only 3-uniform Veblen hypergraph with 4 edges is the 4-uniform simplex (i.e., the tetrahedron) as shown in Cooper and Dutle.
%e There are two 3-uniform Veblen hypergraphs with 5 edges: the crown, {(1,2,3),(1,2,4),(1,2,5),(3,4,5)^2}, and the tight 5-cycle, {(1,2,3),(2,3,4),(3,4,5),(4,5,1),(5,1,2)}.
%o (Sage)
%o e = n
%o #Given a 3-uniform hypergraph H, returns true if H is 3-valent.
%o def is_3_valent(H = IncidenceStructure([[]])):
%o return (Set([H.degree(i) % 3 for i in range(len(H.ground_set()))]) == Set([0]))
%o #Returns a list of all connected 3-uniform Veblen hypergraphs with exactly e edges, up to isomorphism.
%o def Veblen_3graphs(e=1):
%o V = []
%o for n in range(3,e+2): #might be able to give a better bound
%o for H in hypergraphs.nauty(e,n,uniform =3,multiple_sets = True, vertex_min_degree = 3, set_min_size = 3, connected = True):
%o if is_3_valent(IncidenceStructure(H)):
%o V.append(IncidenceStructure(H))
%o return V
%o len(Veblen_3graphs(e))
%K nonn,more
%O 1,5
%A _Gregory J. Clark_, Oct 18 2018
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