Maximum and Minimum Works Performed by T˜n

Abstract

Let Xn and X*n be the finite sets {1,2,3,...,n} and {±1,±2,±3,..,±n} respectively. A map α: Xn→Xn is called a transformation on Xn We call α a signed transformation if α: Xn→X*n Let Tn and T˜n be the sets of full and signed full transformations on Xn respectively. The work, w(α) performed by a transformation α is defined as the sum of all the distances |i-iα| for each i ϵ dom(α) In this paper, we present a range for the values of w(α) for all α ϵ Tn. Further, we characterize elements of T˜n that attain minimum and maximum works and provide formulas for the values of these minimum and maximum.