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The Equivalence Classes of Counting

Author(s)

Haobai Gu

Affiliation(s)

School of Physical Science University of California Irvine, Irvine, CA, US

Abstract

In this paper, we discuss the number of equivalence classes when a permutation group acts on a finite set which consist of mappings. First, we utilize some general permutation subgroups to act on a finite set consists of injective mappings. Next, we extend the case of injective mappings to all mappings from a finite set to a finite set. Moreover, we show the case when some general permutation subgroups act on a finite set consists of some special mappings from a finite set to a finite set. Finally, we give some applications of this topic.

Keywords

Equivalence Class; Permutation Group; Group Action; Mappings From a Finite Set To a Finite Set

References

[1] M. Artin. Algebra. Prentice-Hall, New Jersey, 1991. [2] M. Hall. Combinatorial Theory. John Wiley & Sons, 1986. [3] T. Hungerford. Algebra. Springer-Verlag, New York, 1974. [4] J. H. van Lint, R. M. Wilson. A course in combinatorics. Cambridge University Press, Cambridge, 2001