A058881 Erroneous version of A051504.
1, 1, 7, 137
Offset: 1
Keywords
References
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 220.
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The four symmetry types of the digraphs with 3 nodes are represented by:
1.) {}, the empty graph, has together with the full graph the automorphism group S_3 (as subgroup of S_3) as symmetry type.
2.) {(1,2)} has together with 6 other digraphs the trivial automorphism group {id} as symmetry type. This digraph class is called asymmetric. Their values are given by A051504.
3.) {(1,2),(2,1)} has together with 5 other digraphs the automorphism group containing id and a transposition (so it is C_2 as the subgroup of S_3) as symmetry type.
4.) {(1,2),(2,3),(3,1)} has as the only digraph with three nodes the automorphism group C_3 as symmetry type. As a consequence it has to be self-complementary.
The total of the sizes of the symmetry type classes yields the number of digraphs A000273. Here: 2+7+6+1 = 16 = A000273(3).
Note, that for n > 3 there may be different symmetry types with isomorphic automorphism groups. For n=4 both {(1,2)} and {(1,2),(3,4)} have C_2 as automorphism group, but they are different as permutation group.
For n=4 nodes from the 218 (=A000273(4)) unlabeled nonisomorphic digraphs 136 (=A051504(4)) are asymmetric, so 218 - 136 = 82 are somehow symmetric.
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