Abstract
By using Pólya′s theorem of enumeration and de Bruijn′s generalization of Pólya′s theorem, we obtain the numbers of various weak equivalence classes of functions in RD relative to permutation groups G and H where RD is the set of all functions from a finite set D to a finite set R, G acts on D and H acts on R. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn′s theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non‐isomorphic fm‐graphs relative to a given group.
Highlights
Motivated by Carlitz’s work in [i] on the invariantive properties over a finite field K, Cavior ([2],[3]) and Mullen ([4],[5],[6],[7]) studied several families ofC.Y
Equivalence relations of functions from K into K. These equivalence relations can be described in more general forms as follows: Let D-- {l,2,...,m}, R- {1,2 n}, RD be the set of all functions from D into R, G be a permutation group acting on D and H be a permutation group acting on R
We shall present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn’s theorem, i.e., to determine which functions belong to the same equivalence class
Summary
Motivated by Carlitz’s work in [i] on the invariantive properties over a finite field K, Cavior ([2],[3]) and Mullen ([4],[5],[6],[7]) studied several families of. With the help of Plya’s and de BrulJn’s theorems, our algorithm enables us to determine the numbers of strong equivalence classes relative to some subgroups of the symmetric groups. Let DD be the set of all functions from a finite set D whose cardlnallty is m into itself, G be a permutation group acting on D, and a relation "D be defined on D such that for every f and g e DD f g if and only if there exists a o e G with o-I f(od) g(d) for every d e D. Two matrices A and B are said to be G*-H*related if and only if there exist a P e G* and a Q e H* such that PAQ-I B This relation is an equivalence relation called a G*-H*-relation, and I is partitioned into disjoint equivalence classes each of which is called a G*-H*-.
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