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Self-Complementary Generalized Orbits of a Permutation Group

Published online by Cambridge University Press:  20 November 2018

Roberto Frucht  and
Frank Harary
Roberto Frucht
Affiliation:
Untversidad Técnica Federico Santa Maria, Valparaiso, Chile
Frank Harary
Affiliation:
The University of Michigan, Ann Arbor, U.S.A.
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Abstract

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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 203 - 208
Copyright
Copyright © Canadian Mathematical Society 1974

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