The Identity Transform of a Permutation and its Applications

Abstract

Starting from a Theorem by Hall, we define the identity transform of a permutation π as C(π) = (0 + π(0), 1 + π(1), ..., (n − 1) + π(n − 1)), and we define the set C_n = {(C(π) : π ∈ S_n }, where S_n is the set of permutations of the elements of the cyclic group ℤ _n . In the first part of this paper we study the set C_n : we show some closure properties of this set, and then provide some of its combinatorial and algebraic characterizations and connections with other combinatorial structures. In the second part of the paper, we use some of the combinatorial properties we have determined to provide a different algorithm for the proof of Hall’s Theorem.