Geodesic
Self-Complementary Generalized Orbits of a Permutation Group
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 203-208

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DOI

A permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits. 
Frucht, Roberto; Harary, Frank. Self-Complementary Generalized Orbits of a Permutation Group. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 203-208. doi: 10.4153/CMB-1974-041-6
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     title = {Self-Complementary {Generalized} {Orbits} of a {Permutation} {Group}},
     journal = {Canadian mathematical bulletin},
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     year = {1974},
     volume = {17},
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     doi = {10.4153/CMB-1974-041-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-041-6/}
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