Some combinatorial properties of transformations and their connections with the theory of graphs

Although many results concerning permutations and permutation groups are known, less attention has been paid to transformations and transformation semigroups. It is true that every abstract group is isomorphic to a permutation group, so that with respect to structure there is not difference between abstract groups and permutation groups. Similarly every abstract semigroup is isomorphic to a transformation semigroup; this has led the author to write some papers on the subject [4, 5]. We restrict ourselves to the finite case, and the aim of this paper is to obtain results in this field by a one-to-one correspondence between transformations and directed graphs. The main results of this paper are as follows: (1) Generalization of the Cauchy formula concerning the number of permutations of degree n with prescribed lengths of cycles. (2) Determination of the number of element triples which are generating systems of the symmetric semigroup of degree n, i.e., the semigroup containing every transformation of degree n. (3) Determination of the expected value of the degree of the main permutation of a random transformation of degree n.