Maximum Works Performed by Signed Partial Transformations of a Finite Set - Tal, Pokalas P.; Chibueze, Eze; Sanda, Yulari (Darul Yasin Al Sys, 2025)
Let Xn and Xn* be the finite sets 1, 2, 3, ..., n and ±1, ±2, ±3, ..., ±n respectively. A map from Xn to Xn is called a transformation on Xn. We call a map a signed transformation if it maps from Xn to Xn*. Let Pn be the set of all signed partial transformations on Xn. This set consists of all transformations in Pn for which the domain of the transformation is a subset of Xn. The work w(alpha) performed by a transformation alpha is defined as the sum of all distances |i - alpha(i)| for each i in the domain of alpha. In this paper, we characterize all transformations in Pn that attain maximum and minimum works, and we deduce formulas for these minimum and maximum values. We further present a range for the values of w(alpha) for all transformations in Pn .